Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Richard Andrew Holley

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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

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

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.