Celebratio Mathematica — Liggett — Complete Bibliography

[1] T. M. Liggett: “An invariance principle for conditioned sums of independent random variables,” J. Math. Mech. 18 : 6 (1968), pp. 559–570. MR 238373 Zbl 0181.20502 article

[2] T. M. Liggett and S. A. Lippman: “Stochastic games with perfect information and time average payoff,” SIAM Rev. 11 : 4 (1969), pp. 604–607. MR 260435 Zbl 0193.19602 article

[3] T. M. Liggett: Weak convergence of conditioned sums of independent random vectors. Ph.D. thesis, Stanford University, 1969. Advised by S. Karlin. An article based on this was published in Trans. Am. Math. Soc. 152:1 (1970). MR 2618611 phdthesis

[4] T. M. Liggett: “Convergence of sums of random variables conditioned on a future change of sign,” Ann. Math. Stat. 41 : 6 (1970), pp. 1978–1982. MR 267630 Zbl 0219.60021 article

[5] T. M. Liggett: “Weak convergence of conditioned sums of independent random vectors,” Trans. Am. Math. Soc. 152 : 1 (November 1970), pp. 195–213. Based on the author’s 1969 PhD thesis. MR 268940 Zbl 0221.60018 article

Conditions are given for the weak convergence of processes of the form \[ (\mathbf{X}_n(t) \mid \mathbf{X}_n(1) \in E^n) \] to tied-down stable processes, where \( \mathbf{X}_n(t) \) is constructed from normalized partial sums of independent and identically distributed random vectors which are in the domain of attraction of a multidimensional stable law. The conditioning events are defined in terms of subsets \( E^n \) of \( \mathbb{R}^d \) which converge in an appropriate sense to a set of measure zero. Assumptions which the sets \( E^n \) must satisfy include that they can be expressed as disjoint unions of “asymptotically convex” sets. The assumptions are seen to hold automatically in the special case in which \( E^n \) is taken to be a “natural” neighborhood of a smooth compact hypersurface in \( \mathbb{R}^d \).

[6] T. M. Liggett: “On convergent diffusions: The densities and the conditioned processes,” Indiana Univ. Math. J. 20 : 3 (1970–1971), pp. 265–279. MR 272062 Zbl 0181.44302 article

[7] M. G. Crandall and T. M. Liggett: “Generation of semi-groups of nonlinear transformations on general Banach spaces,” Am. J. Math. 93 : 2 (April 1971), pp. 265–298. MR 287357 Zbl 0226.47038 article

[8] M. G. Crandall and T. M. Liggett: “A theorem and a counterexample in the theory of semigroups of nonlinear transformations,” Trans. Am. Math. Soc. 160 (October 1971), pp. 263–278. MR 301592 Zbl 0226.47037 article

[9] T. M. Liggett: “Existence theorems for infinite particle systems,” Trans. Am. Math. Soc. 165 (1972), pp. 471–481. MR 309218 Zbl 0239.60072 article

[10] D. A. Darling, T. Liggett, and H. M. Taylor: “Optimal stopping for partial sums,” Ann. Math. Stat. 43 : 4 (1972), pp. 1363–1368. MR 312564 Zbl 0244.60037 article

Howard M. Taylor, III	Related
Donald A. Darling	Related

[11] T. M. Liggett: “A characterization of the invariant measures for an infinite particle system with interactions,” Trans. Am. Math. Soc. 179 (1973), pp. 433–453. MR 326867 Zbl 0268.60090 article

[12] T. M. Liggett: “An infinite particle system with zero range interactions,” Ann. Probab. 1 : 2 (1973), pp. 240–253. MR 381039 Zbl 0264.60083 article

[13] T. M. Liggett: “Convergence to total occupancy in an infinite particle system with interactions,” Ann. Probab. 2 : 6 (1974), pp. 989–998. MR 362564 Zbl 0295.60086 article

[14] T. M. Liggett: “A characterization of the invariant measures for an infinite particle system with interactions, II,” Trans. Am. Math. Soc. 198 (1974), pp. 201–213. MR 375531 Zbl 0364.60118 article

[15] R. A. Holley and T. M. Liggett: “Ergodic theorems for weakly interacting infinite systems and the voter model,” Ann. Probab. 3 : 4 (1975), pp. 643–663. MR 402985 Zbl 0367.60115 article

[16] T. M. Liggett: “Ergodic theorems for the asymmetric simple exclusion process,” Trans. Am. Math. Soc. 213 (1975), pp. 237–261. Part II was published in Ann. Probab 5:5 (1977). MR 410986 Zbl 0322.60086 article

Consider the infinite particle system on the integers with the simple exclusion interaction and one-particle motion determined by \[ p(x,x + 1) = p \quad\text{and}\quad p(x,x - 1) = q \] for \( x\in \mathbb{Z} \), where \( p + q = 1 \) and \( p > q \). If \( \mu \) is the initial distribution of the system, let \( \mu_t \) be the distribution at time \( t \). The main results determine the limiting behavior of \( \mu_t \) as \( t\to\infty \) for simple choices of \( \mu \). For example, it is shown that if \( \mu \) is the pointmass on the configuration in which all sites to the left of the origin are occupied, while those to the right are vacant, then the system converges as \( t\to\infty \) to the product measure on \( \{0,1\}^\mathbb{Z} \) with density \( 1/2 \). For the proof, an auxiliary process is introduced which is of interest in its own right. It is a process on the positive integers in which particles move according to the simple exclusion process, but with the additional feature that there can be creation and destruction of particles at one. Ergodic theorems are proved for this process also.

[17] T. M. Liggett: “Coupling the simple exclusion process,” Ann. Probab. 4 : 3 (1976), pp. 339–356. MR 418291 Zbl 0339.60091 article

Consider the infinite particle system on the countable set \( S \) with the simple exclusion interaction and one-particle motion determined by the stochastic transition matrix \( p(x,y) \). In the past, the ergodic theory of this process has been treated successfully only when \( p(x,y) \) is symmetric, in which case great simplifications occur. In this paper, coupling techniques are used to give a complete description of the set of invariant measures for the system in the following three cases:

\( p(x,y) \) is translation invariant on the integers and has mean zero,
\( p(x,y) \) corresponds to a birth and death chain on the nonnegative integers, and
\( p(x,y) \) corresponds to the asymmetric simple random walk on the integers.

[18] T. M. Liggett: “Extensions of the Erdős–Ko–Rado theorem and a statistical application,” J. Comb. Theory, Ser. A 23 : 1 (July 1977), pp. 15–21. MR 441750 Zbl 0361.05011 article

[19] J. Hoffmann-Jørgensen, T. M. Liggett, and J. Neveu: École d’été de probabilités Saint-Flour VI [6th Saint-Flour summer school on probability] (Saint-Flour, France, 22 August–8 September 1976). Edited by P.-L. Hennequin. Lecture Notes in Mathematics 598. Springer (Berlin), 1977. Liggett’s contribution was republished in Interacting particle systems at Saint-Flour (2012). MR 443008 Zbl 0348.00009 book

Paul-Louis Hennequin	Related
Jørgen Hoffmann-Jørgensen	Related
Jacques E. Neveu	Related

[20] T. M. Liggett: “Ergodic theorems for the asymmetric simple exclusion process, II,” Ann. Probab. 5 : 5 (1977), pp. 795–801. Part I was published in Trans. Am. Math. Soc. 213 (1975). MR 445644 Zbl 0378.60104 article

[21] T. M. Liggett: “The stochastic evolution of infinite systems of interacting particles,” pp. 187–248 in J. Hoffmann-Jørgensen, T. M. Liggett, and J. Neveu: École d’été de probabilités Saint-Flour VI [6th Saint-Flour summer school on probability] (Saint-Flour, France, 22 August–8 September 1976). Edited by P.-L. Hennequin. Lecture Notes in Mathematics 598. Springer (Berlin), 1977. MR 458647 Zbl 0363.60109 incollection

Classical statistical mechanics is concerned with the equilibrium theory of certain physical systems. During the past six or eight years, several classes of Markov processes have been proposed as models for the temporal evolution of such systems; and a significant amount of progress has been made in their study. Some of these models, in turn, have been given economic or sociological interpretations. These lectures are intended as an introduction to and survey of some of this recent work. We will say very little about statistical mechanics itself, and will treat our subject as a self-contained branch of probability theory. The reader who would like to know more about the statistical mechanics which lies in the background is referred to [Kinderman and Snell 1980; Preston 1974; Ruelle 1969; Spitzer 1974].

Paul-Louis Hennequin	Related
Jørgen Hoffmann-Jørgensen	Related
Jacques E. Neveu	Related

[22] R. Holley and T. M. Liggett: “The survival of contact processes,” Ann. Probab. 6 : 2 (1978), pp. 198–206. MR 488379 Zbl 0375.60111 article

[23] T. M. Liggett: “Attractive nearest neighbor spin systems on the integers,” Ann. Probab. 6 : 4 (1978), pp. 629–636. MR 494569 Zbl 0381.60095 article

[24] T. M. Liggett: “Random invariant measures for Markov chains and independent particle systems,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 45 : 4 (1978), pp. 297–313. MR 511776 Zbl 0373.60076 article

Let \( P \) be the transition operator for a discrete time Markov chain on a space \( S \). The object of the paper is to study the class of random measures on \( S \) which have the property that \( MP = M \) in distribution. These will be called random invariant measures for \( P \). In particular, it is shown that \( MP = M \) in distribution implies \( MP = M \) a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to \( P \). Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.

[25] T. M. Liggett: “Long range exclusion processes,” Ann. Probab. 8 : 5 (1980), pp. 861–889. MR 586773 Zbl 0457.60079 article

Let \( S \) be a countable set and \( p(x,y) \) be the transition probabilities for a discrete time Markov chain on \( S \). Consider the motion of particles on \( S \) which obey the following rules:

there is always at most one particle at each site in \( S \),
particles wait independent exponential times with mean one before moving, and
when a particle at \( x \) is to move, it moves to \( X_{\tau} \), where \( \{X_n\} \) is the Markov chain starting at \( x \) with transition probabilities \( p(x,y) \) and \( \tau \) is the first time that \( X_n = x \) or \( X_n \) is an unoccupied site.

This process was introduced by Spitzer, and will be called a long range exclusion process because particles may travel long distances in short times. The process is well defined for finite configurations, and we will show how to use monotonicity arguments to define it for arbitrary configurations. It is shown that the configuration in which all sites are occupied may or may not be absorbing for the process. It always is if \( p(x,y) \) is translation invariant on \( S = \mathbb{Z}^d \), but if \( p(x,y) \) is a birth and death chain on \( S = \{0,1 \), \( 2,\dots\} \), it is absorbing if and only if \( p(x,y) \) is recurrent. For each positive function \( \pi(x) \) on \( S \) such that \( \pi P = \pi \), there is a product measure \( \nu_{\pi} \) on \( \{0,1\}^S \) which is a natural candidate for an invariant measure for the process. When \( p(x,y) \) is translation invariant on \( \mathbb{Z}^d \), it is probably the case that \( \nu_{\pi} \) is in fact invariant if and only if \( \pi \) is constant. This will be verified under a mild regularity assumption, which is automatically satisfied if \( d=1 \) or 2 or if the Laplace transform of \( p(0,x) \) is finite in a neighborhood of the origin.

[26] T. M. Liggett: “Interacting Markov processes,” pp. 145–156 in Biological growth and spread: Mathematical theories and applications (Heidelberg, Germany, 16–21 July 1979). Edited by W. Jäger, H. Rost, and P. Tăutu. Lecture Notes in Biomathematics 38. Springer (Berlin), 1980. MR 609355 Zbl 0457.60080 incollection

Hermann Rost	Related
Willi Jäger	Related
Petre Tăutu	Related

[27] R. Durrett and T. M. Liggett: “The shape of the limit set in Richardson’s growth model,” Ann. Probab. 9 : 2 (1981), pp. 186–193. MR 606981 Zbl 0457.60083 article

Let \( C_p \) be the limiting shape of Richardson’s growth model with parameter \( p\in (0,1] \). Our main result is that if \( p \) is sufficiently close to one, then \( C_p \) has a flat edge. This means that \[ \partial C_p \cap \{x\in \mathbb{R}^2: x_1 + x_2 = 1 \} \] is a nondegenerate interval. The value of \( p \) at which this first occurs is shown to be equal to the critical probability for a related contact process. For \( p < 1 \), we show that \( C_p \) is not the full diamond \[ \{x\in \mathbb{R}^2:\|x\| = |x_1| + |x_2| \leq 1\} .\] We also show that \( C_p \) is a continuous function of \( p \), and that when properly rescaled, \( C_p \) converges as \( p\to 0 \) to the limiting shape for exponential site percolation.

[28] R. Holley and T. M. Liggett: “Generalized potlatch and smoothing processes,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 : 2 (1981), pp. 165–195. MR 608015 Zbl 0441.60096 article

We consider simple generalizations of the potlatch and smoothing processes which were introduced in [Spitzer 1981] and studied in [Liggett and Spitzer 1981]. These generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition. The original potlatch and smoothing processes do not exhibit phase transition. Our results show that for the generalized processes, phase transition does not usually occur in one or two dimensions, but usually does occur in higher dimensions. Upper and lower bounds for the relevant critical values are obtained. As one application of our results, we obtain the limiting behavior of the critical values for the linear contact process in \( d \) dimensions as \( d\to\infty \) thus answering a question we raised in [Holley and Liggett 1978]. This is done by comparing the contact process with an appropriate generalized smoothing process.

[29] T. M. Liggett and F. Spitzer: “Ergodic theorems for coupled random walks and other systems with locally interacting components,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 56 (1981), pp. 443–468. MR 621659 Zbl 0444.60096 article

In [1981] the second author introduced a variety of new infinite systems with locally interacting components. On the basis of computations for the finite analogues of these systems, he made conjectures ragarding their limiting behavior as \( t\to \infty \). This paper is devoted to the construction of these processes and to the proofs of these conjectures. We restrict ourselves primarily to spatially homogeneous situations; interesting problems remain unsolved in inhomogeneous cases. Two features distinguish these processes from most other infinite particle systems which have been studied. One is that the state spaces of these systems are noncompact; the other that even though the invariant measures are not generally of product form, one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium. The second moment computations are of inherent interest of course, and they play an important role in the proofs of the ergodic theorems as well.

[30] D. Griffeath and T. M. Liggett: “Critical phenomena for Spitzer’s reversible nearest particle systems,” Ann. Probab. 10 : 4 (1982), pp. 881–895. MR 672290 Zbl 0498.60090 article

[31] T. M. Liggett: “Attractive nearest particle systems,” Ann. Probab. 11 : 1 (1983), pp. 16–33. MR 682797 Zbl 0508.60081 article

We consider certain Markov processes with state space \( \{0,1\}^\mathbb{Z} \) which were introduced and first studied by Spitzer. In these systems, deaths occur at rate one independently of the configuration, and births occur at rate \( \beta(\ell,r) \) where \( \ell \) and \( r \) are the distances to the nearest particles to the left and right, respectively. In his paper, Spitzer gave a necessary and sufficient condition for this process to have a reversible invariant measure, and showed that such a measure must be a stationary renewal process. It was that fact which motivated the study of these systems. Assuming that the process is attractive in the sense of Holley, we give conditions under which (a) the pointmass on the configuration “all zeros” is invariant, and (b) the reversible renewal process is the only nontrivial invariant measure which is translation invariant. As an application, these results allow us to determine exactly the values of \( \lambda > 0 \) and \( p > 0 \) for which the process with \[ \beta(\ell,r)= \lambda(1/\ell + 1/r)^p \] is ergodic.

[32] T. M. Liggett: “Two critical exponents for finite reversible nearest particle systems,” Ann. Probab. 11 : 3 (1983), pp. 714–725. MR 704558 Zbl 0527.60093 article

Finite nearest particle systems are certain one parameter families of continuous time Markov chains \( A_t \) whose state space is the collection of all finite subsets of the integers. Points are added to or taken away from \( A_t \) at rates which have a particular form. The empty set is absorbing for these chains. In the reversible case, the parameter \( \lambda \) is normalized to that extinction at the empty set is certain if and only if \( \lambda \leq 1 \). Let \( \sigma(\lambda) \) be the probability of nonextinction starting from a singleton. In a recent paper, Griffeath and Liggett obtained the bounds \[\lambda^{-1}(\lambda - 1)\leq \sigma(\lambda)\leq |\log \lambda^{-1}(\lambda-1)|^{-1} \] for \( \lambda > 1 \), and raised the question of determining the correct asymptotics of \( \sigma(\lambda) \) as \( \lambda \downarrow 1 \). In the present paper, this question is largely answered by showing under a moment assumption that for \( \lambda > 1 \), \( \sigma(\lambda) \) is bounded above by a constant multiple of \( \lambda - 1 \). In the critical case \( \lambda = 1 \), a similar improvement is made on the known bounds on the asymptotics as \( n\to\infty \) of the probability that \( A_t \) is of cardinality at least \( n \) sometime before extinction. Similar results have been conjectured, but remain open problems in nonreversible situations — for example, for the basic one-dimensional contact process.

[33] R. Durrett and T. M. Liggett: “Fixed points of the smoothing transformation,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 64 : 3 (1983), pp. 275–301. MR 716487 Zbl 0506.60097 article

Let \( W_1,\dots,W_N \) be \( N \) nonnegative random variables and let \( \mathfrak{M} \) be the class of all probability measures on \( [0,\infty) \). Define a transformation \( T \) on \( \mathfrak{M} \) by letting \( T\mu \) be the distribution of \[ W_1X_1+\cdots + W_NX_N ,\] where the \( X_i \) are independent random variables with distribution \( \mu \), which are independent of \( W_1,\dots \), \( W_N \) as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for \( T \) to have a nontrivial fixed point of finite mean in the special cases that the \( W_i \) are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some \( \lambda > 1 \), \( EW_i^{\gamma} < \infty \) for all \( i \), we determine exactly when \( T \) has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, \( T \) always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett.

[34] T. M. Liggett: “Finite nearest particle systems,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 68 : 1 (1984), pp. 65–73. MR 767445 Zbl 0557.60087 article

[35] T. M. Liggett: Interacting particle systems. Grundlehren der Mathematischen Wissenschaften 276. Springer (New York), 1985. Reprinted in 2005. A Russian translation was published as Markovskie protsessy s lokal’nym vzaimodejstviem (1989). MR 776231 Zbl 0559.60078 book

[36] T. M. Liggett: “An improved subadditive ergodic theorem,” Ann. Probab. 13 : 4 (1985), pp. 1279–1285. MR 806224 Zbl 0579.60023 article

[37] T. M. Liggett: “Nearest particle systems: Results and open problems,” pp. 200–215 in Stochastic spatial processes: Mathematical theories and biological applications (Heidelberg, Germany, 10–14 September 1984). Edited by P. Tautu. Lecture Notes in Mathematics 1212. Springer (Berlin), 1986. MR 877777 Zbl 0603.60094 incollection

Nearest particle systems are certain continuous time Markov processes on \( X = \{O,1\}^\mathbb{Z} \) where \( \mathbb{Z} \) is the set of integers). Flips occur from zero to one and from one to zero at each site. The characteristic property of these systems is that the rate at which a coordinate flips depends on the rest of the configuration only through the distances to the nearest sites to the right and left which have the value one. These processes were introduced by Spitzer [1977], and have been the subject of a number of papers since then. A substantial theory of these processes has been developed, but a number of open problems still remain. This paper presents an overview of the theory as it stands today, and a description of some of the more important open problems which should be addressed. No detailed proofs will be provided here, but in many cases the main ideas of the proof will be given. Complete proofs can be found in the original papers, or in [Liggett 1985, Chapter VII].

[38] T. M. Liggett: “Applications of the Dirichlet principle to finite reversible nearest particle systems,” Probab. Theory Relat. Fields 74 : 4 (1987), pp. 505–528. MR 876253 Zbl 0589.60081 article

[39] T. M. Liggett: “Reversible growth models on \( \mathbb{Z}^d \): Some examples,” pp. 213–227 in Percolation theory and ergodic theory of infinite particle systems (Minneapolis, MN, 1984–1985). Edited by H. Kesten. IMA Volumes in Mathematics and its Applications 8. Springer (New York), 1987. MR 894550 Zbl 0636.60104 incollection

In a recent paper [Liggett 1987b], a class of reversible growth models on a fairly general set of sites was introduced and studied. These models are generalizations of the finite reversible nearest particle systems on the integers, which have been considered in several papers in recent years [Griffeath and Liggett 1982; Liggett 1987a], for example). The focus of attention in these growth models is the probability of survival of the system. Typically there are natural one parameter families of models, and one wishes to determine the critical value for that parameter, which is the point at which survival with positive probability begins to occur. Once this is done, it is of interest to determine the manner in which the survival probability approaches its limit (which is usually zero) as the parameter approaches the critical value from above.

The purpose of this paper is to apply the results of [Liggett 1987b] to various examples of models on the \( d \)-dimensional integer lattice \( \mathbb{Z}^d \). We begin in Section 2 by reviewing the main results of [Liggett 1987b], as they appear when specialized to \( \mathbb{Z}^d \). The reader is referred to [Liggett 1987b] for the proofs of these results. The later sections deal with various special classes of models in which the birth rates have a specified form. Among these, we will find examples in which the critical value \( \lambda_c \) is zero, and others in which the critical value is positive. One typical feature of the results is that the survival probability decays like a power of \( \lambda - \lambda_c \) as \( \lambda + \lambda_c \) if \( \lambda_c > 0 \). while it decays much faster than any power of \( \lambda \) if \( \lambda_c = 0 \).

[40] T. M. Liggett: “Reversible growth models on symmetric sets,” pp. 275–301 in Probabilistic methods in mathematical physics (Katata and Kyoto, Japan, 20–26 June 1985). Edited by K. Itô and N. Ikeda. Academic Press (Boston), 1987. MR 933828 Zbl 0653.60094 incollection

Nobuyuki Ikeda	Related
Kiyoshi Itō	Related

[41] T. M. Liggett: “Spatial stochastic growth models–survival and critical behavior,” pp. 1032–1041 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 3–11 August 1986), vol. 2. Edited by A. M. Gleason. American Mathematical Society (Providence, RI), 1987. MR 934305 Zbl 0667.92012 incollection

[42] T. M. Liggett and S. C. Port: “Systems of independent Markov chains,” Stochastic Processes Appl. 28 : 1 (1988), pp. 1–22. MR 936369 Zbl 0657.60123 article

[43] E. D. Andjel, M. D. Bramson, and T. M. Liggett: “Shocks in the asymmetric exclusion process,” Probab. Theory Relat. Fields 78 : 2 (June 1988), pp. 231–247. MR 945111 Zbl 0632.60107 article

In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density \( \lambda \) at \( -\infty \) and \( \rho \) at \( +\infty \). Earlier results described the limiting behavior in all cases except for \( 0 < \lambda < 1/2 \), \( \lambda+\rho = 1 \). Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in [1975]. The convergence proof is based on coupling and symmetry considerations.

Enrique Daniel Andjel	Related
Maury D. Bramson	Related

[44] T. Liggett: Markovskie protsessy s lokal’nym vzaimodejstviem [Markov processes with local interaction]. Mir (Moscow), 1989. Edited and with a preface by R. L. Dobrushin. Russian translation of Interacting particle systems (1985),. MR 993073 Zbl 0741.60102 book

[45] T. M. Liggett: “Total positivity and renewal theory,” pp. 141–162 in Probability, statistics, and mathematics: Papers in honor of Samuel Karlin. Edited by T. W. Anderson, K. B. Athreya, and D. L. Iglehart. Academic Press (Boston), 1989. MR 1031283 Zbl 0682.60078 incollection

Suppose that \( f(k) \) is a probability density on the positive integers, and let \( u(k) \) be the corresponding renewal sequence. Kaluza [1928] and de Bruijn and Erdős [1953] proved several results which relate convexity properties of \( f \) to convexity properties of \( u \). We first note that these convexity properties can be formulated in terms of the total positivity of certain orders of the functions \( f(i+j+1) \) and \( u(i+j) \). This observation permits us to prove an infinite collection of implications which contain the Kaluza and de Bruijn and Erdős results as special cases. In our second result, we show how the imposition of a mild total positivity assumption on \( f(k) \) permits one to give a straightforward proof of the fact that \( u(n)-u(n+1) \) is asymptotic to a constant multiple of the tail probabilities of \( f \). Continuous time versions of these results are discussed briefly. This work was motivated by a problem in interacting particle systems.

Theodore W. Anderson	Related
Donald L. Iglehart	Related
Samuel Karlin	Related
Krishna B. Athreya	Related

[46] T. M. Liggett: “Exponential \( L_2 \) convergence of attractive reversible nearest particle systems,” Ann. Probab. 17 : 2 (1989), pp. 403–432. MR 985371 Zbl 0679.60093 article

Nearest particle systems are continuous-time Markov processes on \( \{0,1\}^\mathbb{Z} \) in which particles die at rate 1 and are born at rates which depend on their distances to the nearest particles to the right and left. There is a natural parametrization of these systems with respect to which they exhibit a phase transition. When the process is attractive and reversible, the critical value \( \lambda_c \) above which a nontrivial invariant measure exists can be computed exactly. This invariant measure is the distribution \( \nu \) of a stationary discrete time renewal process. Under a mild regularity assumption, we prove that the following three statements are equivalent: (a) The nearest particle system converges to equilibrium exponentially rapidly in \( L_2(\nu) \). (b) The density of the interarrival times in the renewal process has exponentially decaying tails. (c) The nearest particle system is supercritical in the sense that \( \lambda > \lambda_c \). Under an additional second-moment assumption, we prove that the critical exponent associated with the exponential convergence is 2. The proof of exponential convergence is based on an unusual comparison of the nearest particle system with an infinite system of independent birth and death chains. To carry out this comparison, a new representation is developed for a stationary renewal process with a log-convex renewal sequence in terms of a sequence of i.i.d. random variables.

[47] W.-D. Ding, R. Durrett, and T. M. Liggett: “Ergodicity of reversible reaction diffusion processes,” Probab. Theory Relat. Fields 85 : 1 (March 1990), pp. 13–26. MR 1044295 Zbl 0669.60077 article

Richard Timothy Durrett	Related
Wan-Ding Ding	Related

[48] T. M. Liggett: “\( L_2 \) rates of convergence for attractive reversible nearest particle systems: The critical case,” Ann. Probab. 19 : 3 (July 1991), pp. 935–959. MR 1112402 Zbl 0737.60092 article

Reversible nearest particle systems are certain one-dimensional interacting particle systems whose transition rates are determined by a probability density \( \beta(n) \) with finite mean on the positive integers. The reversible measure for such a system is the distribution \( \nu \) of the stationary renewal process corresponding to this density. In an earlier paper, we proved under certain regularity conditions that the system converges exponentially rapidly in \( L_2(\nu) \) if and only if the system is supercritical. This in turn is equivalent to \( \beta(n) \) having exponential tails. In this paper, we consider the critical case, and give moment conditions on \( \beta(n) \) which are separately necessary and sufficient for the convergence of the process in \( L_2(\nu) \) at a specified algebraic rate. In order to do so, we develop conditions on the generator of a general Markov process which correspond to algebraic \( L_2 \) convergence of the process. The use of these conditions is also illustrated in the context of birth and death chains on the positive integers.

[49] T. M. Liggett: “Limiting behavior of a one-dimensional system with long range interactions,” pp. 31–40 in Mathematics of random media (Blacksburg, VA, 29 May–9 June 1989). Edited by W. E. Kohler and B. S. White. Lectures in Applied Mathematics 27. American Mathematical Society (Providence, RI), 1991. MR 1117234 Zbl 0731.60094 incollection

During the past twenty years, the study of interacting particle systems has concentrated on certain specific types of models. These have been chosen partly for their simplicity and mathematical appeal, partly because they exhibit phenomena such as phase transition which are of interest in other fields, and partly because they are well suited to analysis by certain important techniques. One type of model which has all of these attributes is known as a nearest particle system. This paper describes the progress which has been made during the past decade in understanding the limiting behavior of this process, with particular emphasis on the most tractable case in which it has the properties of attractiveness and reversibility.

Benjamin Steven White	Related
Werner Erwin Kohler	Related

[50] T. M. Liggett: “Spatially inhomogeneous contact processes,” pp. 105–140 in Spatial stochastic processes. Edited by K. S. Alexander and J. C. Watkins. Progress in Probability 19. Birkäuser (Boston), 1991. A Festschrift in honor of Ted Harris on his seventieth birthday. MR 1144094 Zbl 0747.60096 incollection

A one-dimensional spatially inhomogeneous contact process is a Markov process in \( \{O,1\}^\mathbb{Z} \), where \( \mathbb{Z} \) is the set of integers, which has the following transitions:

\( 1 \to 0 \) at site \( k \) at rate \( \delta(k) \), and
\( 0 \to 1 \) at site \( k \) at rate \( \rho(k)\,\eta(k+1) + \lambda(k)\,\eta(k-1) \),

where \( \delta(k) > 0 \), \( \rho(k)\geq 0 \), \( \lambda(k)\geq 0 \), and \( \eta\in\{O,1\}^\mathbb{Z} \) is the current configuration. If \[ \delta(k)\equiv 1 \quad\text{and}\quad \rho(k)\equiv\eta(k)\equiv\lambda ,\] this is the basic contact process which was first studied by Harris in 1974. If \[ \{(\delta(k),\rho(k),\eta(k))\mid k\in \mathbb{Z}\} \] is chosen randomly in a stationary ergodic manner, it is natural to call this a contact process in a random environment. In this paper, we present three types of results, giving sufficient conditions (a) for extinction of the process, (b) for survival of the process, and (c) for the process to have at most four extremal invariant measures.

Kenneth Sidney Alexander	Related
Theodore Edward Harris	Related
Joseph Clyde Watkins, III	Related

[51] T. M. Liggett: “The periodic threshold contact process,” pp. 339–358 in Random walks, Brownian motion, and interacting particle systems. Edited by R. Durrett and H. Kesten. Progress in Probability 28. Birkhäuser (Boston), 1991. A Festschrift in honor of Frank Spitzer. MR 1146457 Zbl 0747.60097 incollection

Frank Ludvig Spitzer	Related
Richard Timothy Durrett	Related
Harry Kesten	Related

[52] T. M. Liggett: “The survival of one-dimensional contact processes in random environments,” Ann. Probab. 20 : 2 (1992), pp. 696–723. MR 1159569 Zbl 0754.60126 article

Consider the inhomogeneous contact process on \( \mathbb{Z}^1 \) with recovery rate \( \delta(k) \) at site \( k \) and infection rates \( \lambda(k) \) and \( \rho(k) \) at site \( k \) due to the presence of infected neighbors at \( k - 1 \) and \( k + 1 \) respectively. A special case of the main result in this paper is the following: Suppose that the environment is chosen in such a way that the \( \delta(k) \)’s, \( \lambda(k) \)’s and \( \rho(k) \)’s are all mutually independent, with the \( \delta(k) \)’s having a common distribution, and the \( \lambda(k) \)’s and \( \rho(k) \)’s having a common distribution. Then the process survives if \[ E\frac{\delta(\lambda + \rho + \delta)}{\lambda\rho} < 1, \] while the right edge \( r_t \) of the process with initial configuration \[ \cdots 111000\cdots \] satisfies \[ \limsup_{t\to\infty}r_t = +\infty \] if \[ E\log\frac{\delta(\lambda + \rho + \delta)}{\lambda\rho} < 0. \] If the environment is deterministic and periodic with period \( p \), we prove survival if \[ \prod^p_{k=1}\frac{\delta(k)[\lambda(k) + \rho(k-1) + \delta(k)]}{\lambda(k)\rho(k-1)} < 1 \] and \[ \prod^p_{k=1}\frac{\delta(k-1)[\lambda(k) + \rho(k-1) + \delta(k-1)]}{\lambda(k)\rho(k-1)} < 1. \]

[53] J. E. Cohen and T. M. Liggett: “Random arithmetic-geometric means and random pi: Observations and conjectures,” Stochastic Processes Appl. 41 : 2 (June 1992), pp. 261–271. MR 1164179 Zbl 0756.60056 article

[54] E. D. Andjel, T. M. Liggett, and T. Mountford: “Clustering in one-dimensional threshold voter models,” Stochastic Processes Appl. 42 : 1 (August 1992), pp. 73–90. MR 1172508 Zbl 0752.60086 article

We consider one-dimensional spin systems in which the transition rate is 1 at site \( k \) if there are at least \( N \) sites in \[ \{k-N,\,k-N+1,\dots,k+N-1,\,k+N\} \] at which the ‘opinion’ differs from that at \( k \), and the rate is zero otherwise. We prove that clustering occurs for all \( N\geq 1 \) in the sense that \( P[\eta_t(k)\neq \eta_t(j)] \) tends to zero as \( t \) tends to \( \infty \) for every initial configuration. Furthermore, the limiting distribution as \( t\to\infty \) exists (and is a mixture of the pointmasses on \( \eta \equiv 1 \) and \( \eta \equiv 0 \)) if the initial distribution is translation invariant. In case \( N = 1 \), the first of these results was proved and a special case of the second was conjectured in a recent paper by Cox and Durrett.

Now let \( D(\rho) \) be the limiting density of 1’s when the initial distribution is the product measure with density \( \rho \). If \( N = 1 \), we show that \( D(\rho) \) is concave on \( [0,12] \), convex on \( [12,1] \), and has derivative 2 at 0. If \( N\geq 2 \), this derivative is zero.

[55] D. Chen and T. M. Liggett: “Finite reversible nearest particle systems in inhomogeneous and random environments,” Ann. Probab. 20 : 1 (January 1992), pp. 152–173. MR 1143416 Zbl 0753.60099 article

[56] T. M. Liggett: “The coupling technique in interacting particle systems,” pp. 73–83 in Doeblin and modern probability (Blaubeuren, Germany, 2–7 November 1991). Edited by H. Cohn. Contemporary Mathematics 149. American Mathematical Society (Providence, RI), 1993. MR 1229954 Zbl 0796.60099 incollection

Harry Cohn	Related
Wolfgang Doeblin	Related

[57] T. M. Liggett: “Clustering and coexistence in threshold voter models,” pp. 403–410 in Cellular automata and cooperative systems (Les Houches, France, 22 June–2 July 1992). Edited by N. Boccara, E. Goles, S. Martínez, and P. Picco. NATO Science Series C: Mathematical and Physical Sciences 396. Kluwer Academic (Dordrecht), 1993. MR 1267982 Zbl 0869.60086 incollection

Nino Boccara	Related
Eric Goles Chacc	Related
Pierre Picco	Related
Servet A. Martínez	Related

[58] T. M. Liggett: “Survival and coexistence in interacting particle systems,” pp. 209–226 in Probability and phase transition (Cambridge, UK, 4–16 July 1993). Edited by G. Grimmett. NATO Science Series C: Mathematical and Physical Sciences 420. Kluwer Academic (Dordrecht), 1994. MR 1283183 Zbl 0832.60094 incollection

[59] J. Aaronson, T. Liggett, and P. Picco: “Equivalence of renewal sequences and isomorphism of random walks,” Isr. J. Math. 87 : 1–3 (1994), pp. 65–76. MR 1286815 Zbl 0808.60063 article

Pierre Picco	Related
Jonathan Aaronson	Related

[60] G. A. Kriegsmann, K. L. Lange, and T. M. Liggett: “Charles G. Lange (March 30, 1942–June 25, 1993),” Methods Appl. Anal. 1 : 4 (1994), pp. 387–391. MR 1317018 Zbl 0832.01014 article

Charles Gene Lange	Related
Kenneth Lamar Lange	Related
Gregory Anthony Kriegsmann	Related

[61] T. M. Liggett and P. Petersen: “The law of large numbers and \( \sqrt{2} \),” Am. Math. Monthly 102 : 1 (January 1995), pp. 31–35. MR 1321453 Zbl 0823.60023 article

[62] T. M. Liggett: “Improved upper bounds for the contact process critical value,” Ann. Probab. 23 : 2 (1995), pp. 697–723. MR 1334167 Zbl 0832.60093 article

[63] P. A. Ferrari, A. Galves, and T. M. Liggett: “Exponential waiting time for filling a large interval in the symmetric simple exclusion process,” Ann. Inst. H. Poincaré Probab. Statist. 31 : 1 (1995), pp. 155–175. With French abstract. MR 1340035 Zbl 0819.60095 article

Antonio Galves	Related
Pablo Augusto Ferrari	Related

[64] T. M. Liggett: “Survival of discrete time growth models, with applications to oriented percolation,” Ann. Appl. Probab. 5 : 3 (1995), pp. 613–636. MR 1359822 Zbl 0842.60090 article

We prove survival for a class of discrete time Markov processes whose states are finite sets of integers. As applications, we obtain upper bounds for the critical values of various two-dimensional oriented percolation models. The technique of proof is based generally on that used by Holley and Liggett to prove survival of the one-dimensional basic contact process. However, the fact that our processes evolve in discrete time requires that we make substantial changes in the way this technique is used. When applied to oriented percolation on the two-dimensional square lattice, our result gives the following bounds: \( p_c \leq 3/4 \) for bond percolation and \( p_c \leq 3/4 \) for site percolation.

[65] T. M. Liggett: “Multiple transition points for the contact process on the binary tree,” Ann. Probab. 24 : 4 (1996), pp. 1675–1710. MR 1415225 Zbl 0871.60087 article

The contact process on \( \mathbb{Z}^d \) is known to have only two fundamental types of behavior: survival and extinction. Recently Pemantle discovered that the phase structure for the contact process on a tree can be more complex. There are three possible types of behavior: strong survival, weak survival and extinction. He proved that all three occur on homogeneous trees in which each vertex has \( d+1 \) neighbors, provided that \( d\geq 3 \), but he left open the case \( d=2 \). Since \( d=1 \) corresponds to \( \mathbb{Z}^1 \), in which weak survival does not occur, \( d=2 \) is the boundary case. In this paper, we complete this picture, by showing that weak survival does occur on the binary tree for appropriate parameter values. In doing so, we extend and develop techniques for obtaining upper and lower bounds for the critical values associated with strong and weak survival of the contact process on more general graphs.

[66] T. M. Liggett: “Branching random walks and contact processes on homogeneous trees,” Probab. Theory Related Fields 106 : 4 (1996), pp. 495–519. MR 1421990 Zbl 0867.60092 article

Branching random walks and contact processes on the homogeneous tree in which each site has \( d+1 \) neighbors have three possible types of behavior (for \( d\geq 2 \)): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that if there is local extinction, then the probability that an individual ever has a descendent at a site \( n \) units away from that individual’s location is at most \( d^{-n/2} \), while if there is global extinction, this probability is at most \( d^{-n} \). Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and leave others open. In particular, we prove that for some values of the infection parameter \( \lambda \), there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different from those constructed by Durrett and Schinazi in a recent paper.

[67] T. M. Liggett: “Stochastic models of interacting systems,” Ann. Probab. 25 : 1 (1997), pp. 1–29. From the 1996 Wald Memorial Lectures. MR 1428497 Zbl 0873.60072 article

Interacting particle systems is by now a mature area of probability theory, but one that is still very active. We begin this paper by explaining how models from this area arise in fields such as physics and biology. We turn then to a discussion of both older and more recent results about them, concentrating on contact processes, voter models, and exclusion processes. These processes are among the most studied in the field, and have the virtue of relative simplicity in their description, which permits us to address the fundamental issues about their behavior without dealing with the extra complications that models from specific areas of application would require.

[68] T. M. Liggett, R. H. Schonmann, and A. M. Stacey: “Domination by product measures,” Ann. Probab. 25 : 1 (1997), pp. 71–95. MR 1428500 Zbl 0882.60046 article

We consider families of \( \{0,1\} \)-valued random variables indexed by the vertices of countable graphs with bounded degree. First we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site, the probability of seeing a 1 at this site is at least a value \( p \) which is large enough, then this random field dominates a product measure with positive density. Moreover the density of this dominated product measure can be made arbitrarily close to 1, provided that \( p \) is close enough to 1. Next we address the issue of obtaining the critical value of \( p \), defined as the threshold above which the domination by positive-density product measures is assured. For the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than \( k \) units, this critical value is shown to be \[ 1-k^k/(k+1)^{k+1} ,\] and a discontinuous transition is shown to occur. Similar critical values of \( p \) are found for other classes of probability measures on \( \{0,1\}^{\mathbb{Z}} \). For the class of \( k \)-dependent measures the critical value is again \[ 1-k^k/(k+1)^{k+1} ,\] with a discontinuous transition. For the class of two-block factors the critical value is shown to be \( 1/2 \) and a continuous transition is shown to take place in this case. Thus both the critical value and the nature of the transition are different in the two-block factor and 1-dependent cases.

Alan Martin Stacey	Related
Roberto H. Schonmann	Related

[69] V. Belitsky, P. A. Ferrari, N. Konno, and T. M. Liggett: “A strong correlation inequality for contact processes and oriented percolation,” Stochastic Process. Appl. 67 : 2 (1997), pp. 213–225. MR 1449832 Zbl 0890.60094 article

Vladimir Belitsky	Related
Pablo Augusto Ferrari	Related
Norio Konno	Related

[70] T. M. Liggett: “Ultra logconcave sequences and negative dependence,” J. Combin. Theory Ser. A 79 : 2 (August 1997), pp. 315–325. MR 1462561 Zbl 0888.60013 article

[71] T. M. Liggett: “Correction: ‘Coexistence in threshold voter models’,” Ann. Probab. 27 : 4 (1999), pp. 2120–2121. Correction to an article published in Ann. Probab. 22:2 (1994). MR 1742905 Zbl 0814.60094 article

The threshold voter models considered in this paper are special cases of the nonlinear voter models which were introduced recently by Cox and Durrett. They are spin systems on \( \mathbb{Z}^d \) with transition rates \[c(x,\eta) = \begin{cases} 1, \text{ if there is } a_y \text{ with } \|x-y\| \leq N \text{ and } \eta(x)\neq \eta(y), \\ 0, \text{ otherwise.} \\ \end{cases} \] This system is known to cluster if \( N=d=1 \), and to coexist if \( N\geq 4 \) in one dimension and if \( N \) is reasonably large in other dimensions. Cox and Durrett conjectured that it coexists in all cases except \( N=d=1 \). In this paper, we prove this conjecture. The proof is based on comparisons with threshold contact processes. The hard part of the proof consists of showing that the second nearest neighbor threshold contact process in one dimension with parameter 1 survives. The proof of this result is modeled after the proof by Holley and Liggett that the critical value of the basic contact process in one dimension is at most 2. By comparison with that proof, however, the fact that the interaction is not of nearest neighbor type presents substantial additional difficulties. In fact, part of the proof is computer aided.

[72] T. M. Liggett: “Branching random walks on finite trees,” pp. 315–330 in Perplexing problems in probability: Festschrift in honor of Harry Kesten. Edited by M. Bramson and R. Durrett. Progress in Probability 44. Birkhäuser (Boston), 1999. MR 1703138 Zbl 0948.60097 incollection

Richard Timothy Durrett	Related
Maury D. Bramson	Related
Harry Kesten	Related

[73] T. M. Liggett: Stochastic interacting systems: Contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften 324. Springer (Berlin), 1999. MR 1717346 Zbl 0949.60006 book

[74] T. M. Liggett: “Monotonicity of conditional distributions and growth models on trees,” Ann. Probab. 28 : 4 (2000), pp. 1645–1665. MR 1813837 Zbl 1044.60094 article

We consider a sequence of probability measures \( \nu_n \) obtained by conditioning a random vector \( X = (X_1,\dots,X_d) \) with nonnegative integer valued components on \[ X_1 + \cdots + X_d = n-1 \] and give several sufficient conditions on the distribution of \( X \) for \( \nu_n \) to be stochastically increasing in \( n \). The problem is motivated by an interacting particle system on the homogeneous tree in which each vertex has \( d+1 \) neighbors. This system is a variant of the contact process and was studied recently by A. Puha. She showed that the critical value for this process is \( 1/4 \) if \( d=2 \) and gave a conjectured expression for the critical value for all \( d \). Our results confirm her conjecture, by showing that certain \( \nu_n \)’s defined in terms of \( d \)-ary Catalan numbers are stochastically increasing in \( n \). The proof uses certain combinatorial identities satisfied by the \( d \)-ary Catalan numbers.

[75] A. E. Holroyd and T. M. Liggett: “How to find an extra head: Optimal random shifts of Bernoulli and Poisson random fields,” Ann. Probab. 29 : 4 (2001), pp. 1405–1425. MR 1880225 Zbl 1019.60048 article

We consider the following problem:given an i.i.d. family of Bernoulli random variables indexed by \( \mathbb{Z}^d \), find a random occupied site \( X\in\mathbb{Z}^d \) such that relative to \( X \), the other random variables are still i.i.d. Bernoulli. Results of Thorisson imply that such an \( X \) exists for all \( d \). Liggett proved that for \( d = 1 \), there exists an \( X \) with tails \( P(|X|\geq t) \) of order \( ct^{-1/2} \), but none with finite \( 1/2 \)th moment. We prove that for general \( d \) there exists a solution with tails of order \( ct^{-d/2} \), while for \( d=2 \) there is none with finite first moment. We also prove analogous results for a continuum version of the same problem. Finally we prove a result which strongly suggests that the tail behavior mentioned above is the best possible for all \( d \).

[76] T. M. Liggett: “Tagged particle distributions or how to choose a head at random,” pp. 133–162 in In and out of equilibrium: Probability with a physics flavor (Mambucaba, Brazil, 14–19 August 2000). Edited by V. Sidoravicius. Progress in Probability 51. Birkhäuser (Boston), 2002. MR 1901951 Zbl 1108.60319 incollection

Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed by \( \mathbb{Z}^d \) there exists an occupied site \( X\in \mathbb{Z}^d \) with the property that relative to its location, the other variables are still i.i.d. We raise the question of how large such an \( X \) must be. For \( d = 1 \), we prove that any \( X \) with this property satisfies \( E|X|^{\frac{1}{2}} = \infty \). Moreover, there does exist such an \( X \) with tails \( P(|X|\geq n) \) of order \( Cn^{-\frac{1}{2}} \) so these results are essentially best possible. Analogous results for the Poisson process in one dimension are given. The corresponding problem for stationary ergodic sequences is considered also. This project was motivated by some tagged particle problems.

[77] M. Bramson, T. M. Liggett, and T. Mountford: “Characterization of stationary measures for one-dimensional exclusion processes,” Ann. Probab. 30 : 4 (2002), pp. 1539–1575. MR 1944000 Zbl 1039.60086 article

The product Bernoulli measures \( \nu_{\alpha} \) with densities \( \alpha \), \( \alpha\in [0,1] \), are the extremal translation invariant stationary measures for an exclusion process on \( \mathbb{Z} \) with irreducible random walk kernel \( p(\cdot) \). Stationary measures that are not translation invariant are known to exist for finite range \( p(\cdot) \) with positive mean. These measures have particle densities that tend to 1 as \( x\to\infty \) and tend to 0 as \( x\to -\infty \); the corresponding extremal measures form a one-parameter family and are translates of one another. Here, we show that for an exclusion process where \( p(\cdot) \) is irreducible and has positive mean, there are no other extremal stationary measures. When \[ \sum_{x < 0}x^2p(x) = \infty ,\] we show that any nontranslation invariant stationary measure is not a blocking measure; that is, there are always either an infinite number of particles to the left of any site or an infinite number of empty sites to the right of the site. This contrasts with the case where \( p(\cdot) \) has finite range and the above stationary measures are all blocking measures. We also present two results on the existence of blocking measures when \( p(\cdot) \) has positive mean, and \[ p(y)\leq p(x) \quad\text{and}\quad p(-y)\leq p(-x) \] for \( 1\leq x\leq y \). When the left tail of \( p(\cdot) \) has slightly more than a third moment, stationary blocking measures exist. When \( p(-x)\leq p(x) \) for \( x > 0 \) and \[ \sum_{x < 0}x^2p(x) = \infty ,\] stationary blocking measures also exist.

Maury D. Bramson	Related
Thomas Simon Mountford	Related

[78] T. M. Liggett: “Negative correlations and particle systems,” Markov Process. Related Fields 8 : 4 (2002), pp. 547–564. MR 1957219 Zbl 1021.60084 article

[79] P. Bonacich and T. M. Liggett: “Asymptotics of a matrix valued Markov chain arising in sociology,” Stochastic Process. Appl. 104 : 1 (2003), pp. 155–171. MR 1956477 Zbl 1075.60546 article

We consider a discrete time Markov chain whose state space is the set of all \( N{\times}N \) stochastic matrices with zero diagonal entries. This chain models the evolution of relationships among \( N \) individuals who exchange gifts according to probabilities determined by previous exchanges. We determine the stable equilibria for this chain, and prove convergence to a mixture of these. In particular, we show that for generic initial states, the chain converges to a randomly chosen set of constellations made up of disjoint stars. Each star has a center, which is the recipient of all gifts from the other individuals in that star, while the center distributes his gifts only to members of his own star.

[80] A. E. Holroyd, T. M. Liggett, and D. Romik: “Integrals, partitions, and cellular automata,” Trans. Am. Math. Soc. 356 : 8 (2004), pp. 3349–3368. MR 2052953 Zbl 1095.60003 ArXiv math/0302216 article

Alexander Edward Holroyd	Related
Dan Romik	Related

[81] T. M. Liggett and S. W. W. Rolles: “An infinite stochastic model of social network formation,” Stochastic Process. Appl. 113 : 1 (2004), pp. 65–80. MR 2078537 Zbl 1071.60097 article

[82] T. M. Liggett: “Interacting particle systems — an introduction,” pp. 1–56 in School and conference on probability theory (Trieste, Italy, 13–17 May 2002). Edited by G. F. Lawler. ICTP Lecture Notes 17. Abdus Salam International Center for Theoretical Physics (Trieste), 2004. MR 2198846 Zbl 1221.60141 incollection

Interacting particle systems is a large and growing field of probability theory that is devoted to the rigorous analysis of certain types of models that arise in statistical physics, biology, economics, and other fields. In these notes, we provide an introduction to some of these models, give some basic results about them, and explain how certain important tools are used in their study. The first chapter describes contact, voter and exclusion processes, and introduces the tools of coupling and duality. Chapter 2 is devoted to an analysis of translation invariant linear voter models, using primarily the duality that is available in that case. Chapters 3–5 are concerned with the exclusion process, beginning with the symmetric case, in which one can also use duality, and following with asymmetric systems, which are studied using coupling and other monotonicity techniques. At the end, we report on some very recent work on the stationary distributions of one dimensional systems with positive drift.

[83] T. M. Liggett: Interacting particle systems, Reprinted edition. Classics in Mathematics. Springer (New York), 2005. Reprint of the 1985 original. MR 2108619 Zbl 1103.82016 book

[84] R. Arratia and T. M. Liggett: “How likely is an i.i.d. degree sequence to be graphical?,” Ann. Appl. Probab. 15 : 1B (2005), pp. 652–670. MR 2114985 Zbl 1079.05023 ArXiv math/0504096 article

[85] M. Bramson and T. M. Liggett: “Exclusion processes in higher dimensions: Stationary measures and convergence,” Ann. Probab. 33 : 6 (2005), pp. 2255–2313. MR 2184097 Zbl 1099.60067 article

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on \( \mathbb{Z} \). The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on \( \mathbb{Z}^d \) and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on \( \mathbb{Z}^d \). In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as \( t\to\infty \) for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.

[86] T. M. Liggett and J. E. Steif: “Stochastic domination: The contact process, Ising models and FKG measures,” Ann. Inst. H. Poincaré Probab. Statist. 42 : 2 (March–April 2006), pp. 223–243. MR 2199800 Zbl 1087.60074 ArXiv math/0504530 article

We prove for the contact process on \( \mathbb{Z}^d \), and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate \( \lambda \) is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for “downward” FKG measures. As a consequence of this general result, we show that the plus and minus states for the Ising model on \( \mathbb{Z}^d \) dominate the same set of product measures. We show that this latter fact fails completely on the homogeneous 3-ary tree. We also provide a different distinction between \( \mathbb{Z}^d \) and the homogeneous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on \( \mathbb{Z}^d \) are never stochastically ordered, on the homogeneous 3-ary tree, almost the complete opposite is the case. Next, we show that on \( \mathbb{Z}^d \), the set of product measures which the plus state for the Ising model dominates is strictly increasing in the temperature. Finally, we obtain a necessary and sufficient condition for a finite number of variables, which are both FKG and exchangeable, to dominate a given product measure.

[87] T. M. Liggett: “Conditional association and spin systems,” ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), pp. 1–19. MR 2235171 Zbl 1157.60088 ArXiv math/0507392 article

A 1977 theorem of T. Harris states that an attractive spin system preserves the class of associated probability measures. We study analogues of this result for measures that satisfy various conditional positive correlations properties. In particular, we show that a spin system preserves measures satisfying the FKG lattice condition (essentially) if and only if distinct spins flip independently. The downward FKG property, which has been useful recently in the study of the contact process, lies between the properties of lattice FKG and association. We prove that this property is preserved by a spin system if the death rates are constant and the birth rates are additive (e.g., the contact process), and prove a partial converse to this statement. Finally, we introduce a new property, which we call downward conditional association, which lies between the FKG lattice condition and downward FKG, and find essentially necessary and sufficient conditions for this property to be preserved by a spin system. This suggests that the latter property may be more natural than the downward FKG property.

[88] T. M. Liggett, J. E. Steif, and B. Tóth: “Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem,” Ann. Probab. 35 : 3 (2007), pp. 867–914. MR 2319710 Zbl 1126.44007 ArXiv math/0512191 article

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie–Weiss Ising model and includes as well all ferromagnetic Curie–Weiss Potts and Curie–Weiss Heisenberg models. By de Finetti’s theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that “ferromagnetism” is not however in itself sufficient and also study in some detail the Curie–Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie–Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a “formula” for the extension which is valid in many cases.

Jeffrey Edward Steif	Related
Bálint Tóth	Related

[89] L. Chayes and T. M. Liggett: “One dimensional nearest neighbor exclusion processes in inhomogeneous and random environments,” J. Stat. Phys. 129 : 2 (2007), pp. 193–203. MR 2358802 Zbl 1141.82015 ArXiv math/0701180 article

[90] T. M. Liggett, R. B. Schinazi, and J. Schweinsberg: “A contact process with mutations on a tree,” Stochastic Process. Appl. 118 : 3 (2008), pp. 319–332. MR 2389047 Zbl 1141.60065 ArXiv math/0612564 article

Consider the following stochastic model for immune response. Each pathogen gives birth to a new pathogen at rate \( \lambda \). When a new pathogen is born, it has the same type as its parent with probability \( 1-r \). With probability \( r \), a mutation occurs, and the new pathogen has a different type from all previously observed pathogens. When a new type appears in the population, it survives for an exponential amount of time with mean 1, independently of all the other types. All pathogens of that type are killed simultaneously. Schinazi and Schweinsberg [R.B. Schinazi, J. Schweinsberg, Spatial and non-spatial stochastic models for immune response, Markov Process. Related Fields (2006) (in press)] have shown that this model on \( \mathbb{Z}^d \) behaves rather differently from its non-spatial version. In this paper, we show that this model on a homogeneous tree captures features from both the non-spatial version and the version. We also obtain comparison results, between this model and the basic contact process on general graphs.

Jason Ross Schweinsberg	Related
Rinaldo Bruno Schinazi	Related

[91] J. Borcea, P. Brändén, and T. M. Liggett: “Negative dependence and the geometry of polynomials,” J. Am. Math. Soc. 22 : 2 (2009), pp. 521–567. MR 2476782 Zbl 1206.62096 ArXiv 0707.2340 article

We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures and uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence, and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures, and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.

Petter Brändén	Related
Julius Bogdan Borcea	Related

[92] T. M. Liggett: “Distributional limits for the symmetric exclusion process,” Stochastic Process. Appl. 119 : 1 (January 2009), pp. 1–15. MR 2485017 Zbl 1172.60031 article

[93] T. M. Liggett, S. A. Lippman, and R. P. Rumelt: “The asymptotic Shapley value for a simple market game,” Econom. Theory 40 : 2 (2009), pp. 333–338. MR 2507870 Zbl 1173.91372 article

Steven Arthur Lippman	Related
Richard P. Rumelt	Related

[94] T. M. Liggett and R. B. Schinazi: “A stochastic model for phylogenetic trees,” J. Appl. Probab. 46 : 2 (June 2009), pp. 601–607. MR 2535836 Zbl 1180.60079 article

[95] T. M. Liggett: Continuous time Markov processes: An introduction. Graduate Studies in Mathematics 113. American Mathematical Society (Providence, RI), 2010. MR 2574430 Zbl 1205.60002 book

[96] P. Caputo, T. M. Liggett, and T. Richthammer: “Proof of Aldous’ spectral gap conjecture,” J. Am. Math. Soc. 23 : 3 (2010), pp. 831–851. MR 2629990 Zbl 1203.60145 ArXiv 0906.1238 article

Aldous’ spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices with rows and columns indexed by permutations.

Pietro Caputo	Related
Thomas Richthammer	Related

[97] G. R. Grimmett, T. M. Liggett, and T. Richthammer: “Percolation of arbitrary words in one dimension,” Random Structures Algorithms 37 : 1 (2010), pp. 85–99. MR 2674622 Zbl 1202.60155 ArXiv 0807.1676 article

Geoffrey Richard Grimmet	Related
Thomas Richthammer	Related

[98] T. M. Liggett: “Stochastic models for large interacting systems and related correlation inequalities,” Proc. Natl. Acad. Sci. USA 107 : 38 (2010), pp. 16413–16419. MR 2726545 Zbl 1256.60037 article

A very large and active part of probability theory is concerned with the formulation and analysis of models for the evolution of large systems arising in the sciences, including physics and biology. These models have in their description randomness in the evolution rules, and interactions among various parts of the system. This article describes some of the main models in this area, as well as some of the major results about their behavior that have been obtained during the past 40 years. An important technique in this area, as well as in related parts of physics, is the use of correlation inequalities. These express positive or negative dependence between random quantities related to the model. In some types of models, the underlying dependence is positive, whereas in others it is negative. We give particular attention to these issues, and to applications of these inequalities. Among the applications are central limit theorems that give convergence to a Gaussian distribution.

[99] T. M. Liggett: “T. E. Harris’ contributions to interacting particle systems and percolation,” Ann. Probab. 39 : 2 (2011), pp. 407–416. MR 2789500 Zbl 1213.60159 ArXiv 1103.1988 article

[100] T. M. Liggett and A. Vandenberg-Rodes: “Stability on \( \{0,1,2,\dots\}^S \): Birth-death chains and particle systems,” pp. 311–329 in Notions of positivity and the geometry of polynomials: Dedicated to the memory of Julius Borcea. Edited by P. Brändén, M. Passare, and M. Putinar. Trends in Mathematics. Birkhäuser (Basel), 2011. MR 3051173 Zbl 1278.60144 ArXiv 1009.4899 incollection

Mihai Putinar	Related
Julius Bogdan Borcea	Related
Petter Brändén	Related
Alexander Vandenberg-Rodes	Related
Mikael Passare	Related

[101] R. Durrett, T. M. Liggett, F. Spitzer, and A.-S. Sznitman: Interacting particle systems at Saint-Flour (Saint-Flour, France, 1964–1993). Probability at Saint-Flour. Springer (Berlin), 2012. Reprint of various lectures originally published Lecture Notes in Mathematics 390 (1974), \xlink589 (1977)|MR:0443008, 1464 (1991) and 1608 (1995). MR 3075635 Zbl 1248.82021 book

Frank Ludvig Spitzer	Related
Richard Timothy Durrett	Related
Alain-Sol Sznitman	Related

[102] R. Durrett, T. Liggett, and Y. Zhang: “The contact process with fast voting,” Electron. J. Probab. 19 (2014), pp. Article no. 28, 19 pages. MR 3174840 Zbl 1291.60203 article

Consider a combination of the contact process and the voter model in which deaths occur at rate 1 per site, and across each edge between nearest neighbors births occur at rate \( \lambda \) and voting events occur at rate \( \theta \). We are interested in the asymptotics as \( \theta\to\infty \) of the critical value \( \lambda_c(\theta) \) for the existence of a nontrivial stationary distribution. In \( d\geq 3 \), \[ \lambda_c(\theta)\to 1/(2d\rho_d) \] where \( \rho_d \) is the probability a \( d \) dimensional simple random walk does not return to its starting point. In \( d = 2 \), \[ \lambda_c(\theta)/\log(\theta)\to 1/4\pi ,\] while in \( d=1 \), \( \lambda_c(\theta)/\theta^{1/2} \) has \[ \liminf \geq 1/\sqrt{2} \quad\text{and}\quad \limsup < \infty .\] The lower bound might be the right answer, but proving this, or even getting a reasonable upper bound, seems to be a difficult problem.

Yuan Zhang	Related
Richard Timothy Durrett	Related

[103] R. Arratia, T. M. Liggett, and M. J. Williamson: “Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations,” Electron. Commun. Probab. 19 (2014). Article no. 39, 10 pages. MR 3225870 Zbl 1320.60010 ArXiv 1306.3017 article

In discrete contexts such as the degree distribution for a graph, scale-free has traditionally been defined to be power-law. We propose a reasonable interpretation of scale-free, namely, invariance under the transformation of \( p \)-thinning, followed by conditioning on being positive.

For each \( \beta\in (1,2) \), we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-\( \beta \), and different from the usual Yule–Simon power law-\( \beta \) that arises in preferential attachment models.

In addition to characterizing these fixed points, we prove convergence results for iterates of the transformation.

Malcolm J. Williamson.	Related
Richard Alejandro Arratia	Related

[104] A. E. Holroyd and T. M. Liggett: “Symmetric 1-dependent colorings of the integers,” Electron. Commun. Probab. 20 (2015). Article no. 31, 8 pages. MR 3327870 Zbl 1385.60014 ArXiv 1407.4514 article

[105] A. E. Holroyd and T. M. Liggett: “Finitely dependent coloring,” Forum Math. Pi 4 (2016). Article no. e9, 43 pages; Dedicated to Oded Schramm. MR 3570073 Zbl 1361.60025 ArXiv 1403.2448 article

We prove that proper coloring distinguishes between block factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and asked whether a \( k \)-dependent \( q \)-coloring exists for any \( k \) and \( q \). We give a complete answer by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovász local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving \( d \) dimensions and shifts of finite type; in fact, any nondegenerate shift of finite type also distinguishes between block factors and finitely dependent processes.

Alexander Edward Holroyd	Related
Oded Schramm	Related

[106] S. Ghosh, T. M. Liggett, and R. Pemantle: “Multivariate CLT follows from strong Rayleigh property,” pp. 139–147 in 2017 proceedings of the fourteenth workshop on analytic algorithmics and combinatorics (ANALCO) (Barcelona, 16–17 January 2017). Edited by C. Martínez and M. D. Ward. SIAM (Philadelphia, PA), 2017. MR 3630942 Zbl 1432.60034 incollection

Let \( (X_1,\dots,X_d) \) be a random nonnegative integer vector. Many conditions are known to imply a central limit theorem for a sequence of such random vectors, for example, independence and convergence of the normalized covariances, or various combinatorial conditions allowing the application of Stein’s method, couplings, etc. Here, we prove a central limit theorem directly from hypotheses on the probability generating function \( f(z_1,\dots,z_d) \). In particular, we show that the \( f \) being real stable (meaning no zeros with all coordinates in the open upper half plane) is enough to imply a CLT under a nondegeneracy condition on the variance. Known classes of distributions with real stable generating polynomials include spanning tree measures, conditioned Bernoullis and counts for determinantal point processes. Soshnikov [2002] showed that occupation counts of disjoint sets by a determinantal point process satisfy a multivariate CLT. Our results extend Soshnikov’s to the class of real stable laws. The class of real stable laws is much larger than the class of determinantal laws, being defined by inequalities rather than identities. Along the way we investigate the related problem of stable multiplication.

Mark Daniel Ward	Related
Subhroshekhar Ghosh	Related
Conrado Martínez	Related
Robin Alexander Pemantle	Related

[107] T. M. Liggett and W. Tang: One-dependent colorings of the star graph. Preprint, April 2018. ArXiv 1804.06877 techreport

This paper is concerned with symmetric 1-dependent colorings of the \( d \)-ray star graph \( \mathscr{S}^d \) for each \( d\geq 2 \). We compute the critical point of the 1-dependent hard-core processes on \( \mathscr{S}^d \), which gives a lower bound for the number of colors needed for a 1-dependent coloring of \( \mathscr{S}^d \). We provide an explicit construction of a 1-dependent \( q \)-coloring for any \( q\geq 5 \) of the infinite subgraph \( \mathscr{S}^3_{(1,1,\infty)} \), which is symmetric in the colors and whose restriction to any copy of \( \mathbb{Z} \) is some symmetric 1-dependent \( q \)-coloring of \( \mathbb{Z} \). We also prove that there is no such coloring of \( \mathscr{S}^3_{(1,1,\infty)} \) with \( q=4 \) colors. A list of open problems are presented.

[108]T. M. Liggett: “Approximating multiples of strong Rayleigh random variables,” Celebratio Mathematica (2021). article