arXiv subjects

Thomas M. Liggett

Publications and source records attributed to Thomas M. Liggett.

17 recordsLinked to original sources

arXiv2023

One-dependent colorings of the star graph

This paper is concerned with symmetric $1$-dependent colorings of the $d$-ray star graph $\mathscr{S}^d$ for $d \ge 2$. 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 $q \ge 5$ of the infinite subgraph $\mathscr{S}^3_{(1,1,\infty)}$, which is symmetric in the colors and whose restriction to any path is some symmetric $1$-dependent $q$-coloring. 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.

math.PR↗

arXiv2016

Multivariate CLT follows from strong Rayleigh property

Let $(X_1 , \ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, \ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~\cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.

math.PR↗

arXiv2015

Finitely dependent coloring

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. More precisely, Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and conjectured that no stationary k-dependent q-coloring exists for any k and q. We disprove this by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring, thus resolving the question for all k and q. 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 Lovasz 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 non-degenerate shift of finite type also distinguishes between block-factors and finitely dependent processes.

math.PR↗

arXiv2014

Symmetric 1-Dependent Colorings of the Integers

In a recent paper by the same authors, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary k-dependent q-coloring for any k and q. When the analogous construction is carried out for q>4 colors, the resulting process is not k-dependent for any k. We construct here a process that is symmetric in the colors and 1-dependent for every q>=4. The construction uses a recursion involving Chebyshev polynomials evaluated at $\sqrt{q}/2$.

math.PR↗

arXiv2014

Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations

In discrete contexts such as the degree distribution for a graph, \emph{scale-free} has traditionally been \emph{defined} to be \emph{power-law}. We propose a reasonable interpretation of \emph{scale-free}, namely, invariance under the transformation of $p$-thinning, followed by conditioning on being positive. For each $β\in (1,2)$, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$β$, and different from the usual Yule--Simon power law-$β$ that arises in preferential attachment models. In addition to characterizing these fixed points, we prove convergence results for iterates of the transformation.

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arXiv2011

Stability on {0,1,2,...}^S: birth-death chains and particle systems

A strong negative dependence property for measures on {0,1}^n - stability - was recently developed in [5], by considering the zero set of the probability generating function. We extend this property to the more general setting of reaction-diffusion processes and collections of independent Markov chains. In one dimension the generalized stability property is now independently interesting, and we characterize the birth-death chains preserving it.

math.PR↗

arXiv2011

T. E. Harris' contributions to interacting particle systems and percolation

Interacting particle systems and percolation have been among the most active areas of probability theory over the past half century. Ted Harris played an important role in the early development of both fields. This paper is a bird's eye view of his work in these fields, and of its impact on later research in probability theory and mathematical physics.

math.PR↗

arXiv2009

Proof of Aldous' spectral gap conjecture

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 on permutations.

math.PR↗

arXiv2008

Negative dependence and the geometry of polynomials

We introduce the class of {\em 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, 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.

math.PR↗

arXiv2008

Percolation of arbitrary words in one dimension

We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.

math.PR↗

arXiv2007

Distributional Limits for the Symmetric Exclusion Process

Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and normal distributions for various functionals of the process.

math.PR↗

arXiv2007

Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem

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.

math.PR↗

arXiv2007

One dimensional nearest neighbor exclusion processes in inhomogeneous and random environments

The processes described in the title always have reversible stationary distributions. In this paper, we give sufficient conditions for the existence of, and for the nonexistence of, nonreversible stationary distributions. In the case of an i.i.d. environment, these combine to give a necessary and sufficient condition for the existence of nonreversible stationary distributions.

math.PR↗

arXiv2006

A contact process with mutations on a tree

Consider the following stochastic model for immune response. Each pathogen gives birth to a new pathogen at rate $λ$. 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 (2006) have shown that this model on $\Z^d$ 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 $\Z^d$ version. We also obtain comparison results between this model and the basic contact process on general graphs.

math.PR↗

arXiv2005

Conditional Association and Spin Systems

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.

math.PR↗

arXiv2005

How likely is an i.i.d. degree sequence to be graphical?

Given i.i.d. positive integer valued random variables D_1,...,D_n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D_1,...,D_n. We give sufficient conditions on the distribution of D_i for the probability that this be the case to be asymptotically 0, {1/2} or strictly between 0 and {1/2}. These conditions roughly correspond to whether the limit of nP(D_i\geq n) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs.

math.PR↗

arXiv2003

Integrals, Partitions, and Cellular Automata

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{π^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

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