Celebratio Mathematica

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 \( \lambda_c = 0 \).

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
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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

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Thomas Milton Liggett

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