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
Abstract
People
BibTeX
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.
@article {key606981m,
AUTHOR = {Durrett, Richard and Liggett, Thomas
M.},
TITLE = {The shape of the limit set in {R}ichardson's
growth model},
JOURNAL = {Ann. Probab.},
FJOURNAL = {Annals of Probability},
VOLUME = {9},
NUMBER = {2},
YEAR = {1981},
PAGES = {186--193},
DOI = {10.1214/aop/1176994460},
NOTE = {MR:606981. Zbl:0457.60083.},
ISSN = {0091-1798},
}
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
Abstract
People
BibTeX
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.
@article {key716487m,
AUTHOR = {Durrett, Richard and Liggett, Thomas
M.},
TITLE = {Fixed points of the smoothing transformation},
JOURNAL = {Z. Wahrscheinlichkeitstheor. Verw. Geb.},
FJOURNAL = {Zeitschrift f\"ur Wahrscheinlichkeitstheorie
und Verwandte Gebiete},
VOLUME = {64},
NUMBER = {3},
YEAR = {1983},
PAGES = {275--301},
DOI = {10.1007/BF00532962},
NOTE = {MR:716487. Zbl:0506.60097.},
ISSN = {0044-3719},
}
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
Abstract
People
BibTeX
Reaction-diffusion processes were introduced by Nicolis and Prigogine, and Haken. Existence theorems have been established for most models, but not much is known about ergodic properties. In this paper we study a class of models which have a reversible measure. We show that the stationary distribution is unique and is the limit starting from any initial distribution.
@article {key1044295m,
AUTHOR = {Ding, Wan-Ding and Durrett, Richard
and Liggett, Thomas M.},
TITLE = {Ergodicity of reversible reaction diffusion
processes},
JOURNAL = {Probab. Theory Relat. Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {85},
NUMBER = {1},
MONTH = {March},
YEAR = {1990},
PAGES = {13--26},
DOI = {10.1007/BF01377624},
NOTE = {MR:1044295. Zbl:0669.60077.},
ISSN = {0178-8051},
}
T. M. Liggett :
“The periodic threshold contact process 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
Abstract
People
BibTeX
We consider the periodic threshold contact process with period 2 in one dimension with parameters \( \lambda \) and \( \mu \) . This process dies out if
\[ \lambda + \mu + 2 > 4\lambda\mu .\]
We obtain a sufficient condition for its survival, which is satisfied by \( (\lambda,\mu) = (2.17,2.18) \) , \( (2.00,2.37) \) , and \( (1.50,3.62) \) , for example. These results were motivated by recent work of Cox and Durrett on the threshold voter model.
@incollection {key1146457m,
AUTHOR = {Liggett, Thomas M.},
TITLE = {The periodic threshold contact process},
BOOKTITLE = {Random walks, {B}rownian motion, and
interacting particle systems},
EDITOR = {Durrett, Rick and Kesten, Harry},
SERIES = {Progress in Probability},
NUMBER = {28},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston},
YEAR = {1991},
PAGES = {339--358},
DOI = {10.1007/978-1-4612-0459-6_19},
NOTE = {A Festschrift in honor of Frank Spitzer.
MR:1146457. Zbl:0747.60097.},
ISSN = {1050-6977},
ISBN = {9781461267706},
}
T. M. Liggett :
“Branching random walks on finite trees 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
Abstract
People
BibTeX
@incollection {key1703138m,
AUTHOR = {Liggett, Thomas M.},
TITLE = {Branching random walks on finite trees},
BOOKTITLE = {Perplexing problems in probability:
{F}estschrift in honor of {H}arry {K}esten},
EDITOR = {Bramson, Maury and Durrett, Rick},
SERIES = {Progress in Probability},
NUMBER = {44},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston},
YEAR = {1999},
PAGES = {315--330},
DOI = {10.1007/978-1-4612-2168-5_17},
NOTE = {MR:1703138. Zbl:0948.60097.},
ISSN = {1050-6977},
ISBN = {9780817640934},
}
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
People
BibTeX
@book {key3075635m,
AUTHOR = {Durrett, Rick and Liggett, Thomas M.
and Spitzer, Frank and Sznitman, Alain-Sol},
TITLE = {Interacting particle systems at {S}aint-{F}lour},
SERIES = {Probability at Saint-Flour},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2012},
PAGES = {viii+331},
URL = {https://www.springer.com/gp/book/9783642252976},
NOTE = {(Saint-Flour, France, 1964--1993). Reprint
of various lectures originally published
\textit{Lecture Notes in Mathematics}
\textbf{390} (1974), \xlink{\textbf{589}
(1977)|MR:0443008}, \textbf{1464} (1991)
and \textbf{1608} (1995). MR:3075635.
Zbl:1248.82021.},
ISSN = {2193-648X},
ISBN = {9783642252976},
}
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
Abstract
People
BibTeX
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.
@article {key3174840m,
AUTHOR = {Durrett, Richard and Liggett, Thomas
and Zhang, Yuan},
TITLE = {The contact process with fast voting},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {19},
YEAR = {2014},
PAGES = {Article no. 28, 19 pages},
DOI = {10.1214/EJP.v19-3021},
NOTE = {MR:3174840. Zbl:1291.60203.},
}
