Celebratio Mathematica

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 \( d \).

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

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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

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.

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Thomas Milton Liggett

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