Proof of Aldous' spectral gap conjecture (2009)
Caputo, Pietro, Liggett, Thomas M., Richthammer, Thomas
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...
Percolation of arbitrary words in one dimension (2008)
Grimmett, Geoffrey R., Liggett, Thomas M., Richthammer, Thomas
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...
Abstracts Plenary Session Unitary and automorphic duals (2008)
Jian-shu Li, Thomas M. Liggett
Let G be a semi-simple Lie group. The automorphic dual, ˆ Gaut, is defined to be the set of all irreducible unitary representations of G which “occur ” in L 2 (Γ\G) for some lattice Γ ⊂ G....
Distributional Limits for the Symmetric Exclusion Process (2008)
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 Gaussian...
Percolation of arbitrary words in one dimension (2008)
Geoffrey R. Grimmett, Thomas M. Liggett, Thomas Richthammer
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...
, nd a random occupied site (2007)
Er E. Holroyd, Thomas M. Liggett
We consider the following problem: given an i.i.d. family of Bernoulli random variables indexed by Z d
Tagged particle distributions, or how to choose a head at random (2007)
Abstract. Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed by Z d, there exists an occupied site X 2 Z d with the...
Tagged particle distributions, or how to choose a head at random (2007)
Abstract. Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed by Z d, there exists an occupied site X 2 Z d with the...
CHARACTERIZATION OF STATIONARY MEASURES FOR ONE DIMENSIONAL EXCLUSION PROCESSES (2007)
Maury Bramson, Thomas M. Liggett
Abstract. The product Bernoulli measures with densities , 2 [0; 1], are the extremal translation invariant stationary measures for an exclusion process on Z with irreducible random walk kernel p()....
Negative Correlations And Particle Systems (2007)
We consider the symmetric exclusion process and systems of independent Markov chains. For each of these, we prove that certain classes of distributions with negative dependence are preserved by the...
A Strong Correlation Inequality For Contact Processes And Oriented Percolation (2007)
Vladimir Belitsky, Pablo A. Ferrari, Norio Konno, Thomas M. Liggett
this paper is devoted to the proof of some inequalities which this quantity must satisfy. Then we consider the analogous issue for oriented bond percolation, which can be viewed as a discrete time...
Distributional Limits for the Symmetric Exclusion Process (2007)
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...
Negative dependence and the geometry of polynomials (2007)
Borcea, Julius, Brändén, Petter, Liggett, Thomas M.
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...
One dimensional nearest neighbor exclusion processes in inhomogeneous and random environments (2007)
Chayes, Lincoln, Liggett, Thomas M.
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...
One Dimensional Nearest Neighbor Exclusion (2007)
Lincoln Chayes, Thomas M. Liggett
Abstract 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,...
The Asymptotic Shapley Value for a Simple Market Game (2007)
Thomas M. Liggett, Richard P. Rumelt
file name: ETDec2307.tex
One Dimensional Nearest Neighbor Exclusion (2007)
Lincoln Chayes, Thomas M. Liggett
Abstract. 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,...
DISTRIBUTIONAL LIMITS FOR THE SYMMETRIC EXCLUSION PROCESS (2007)
Abstract. 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...
A contact process with mutations on a tree (2006)
Liggett, Thomas M., Schinazi, Rinaldo B., Schweinsberg, Jason
Consider the following stochastic model for immune response. Each pathogen gives birth to a new pathogen at rate $\lambda$. When a new pathogen is born, it has the same type as its parent with...
Conditional Association and Spin Systems (2006)
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...
A contact process with mutations on a tree (2006)
Thomas M. Liggett, Rinaldo B. Schinazi, Jason Schweinsberg
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...
Liggett, Thomas M., Steif, Jeffrey E., Tóth, Bálint
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...
Conditional Association and Spin Systems (2005)
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...
How likely is an i.i.d. degree sequence to be graphical? (2005)
Arratia, Richard, Liggett, Thomas M.
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...
How likely is an i.i.d. degree sequence to be graphical? (2005)
Arratia, Richard, Liggett, Thomas M.
Given i.i.d. positive integer valued random variables D1,…,Dn, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D1,…,Dn. We give sufficient...
The Asympotic Shapley Value for a Simple Market Game (2005)
Thomas M. Liggett, Richard P. Rumelt
We consider the game in which b buyers seek to purchase 1 unit of an indivisible good from s sellers, each of whom have k units to sell. The good is worth 0 to each seller and 1 to each buyer. Using...
Richard Arratia, Thomas M. Liggett
Given i.i.d. positive integer valued random variables D1 ; :::; Dn , one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D1 ; :::; Dn . We give sucient...
Abstract. We prove for the contact process on Z d, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate λ is...
Integrals, Partitions, and Cellular Automata (2003)
Holroyd, Alexander E., Liggett, Thomas M., Romik, Dan
We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0
Integrals, Partitions, and Cellular Automata (2003)
Alexander E. Holroyd, Thomas M. Liggett, Dan Romik
We prove that x where f(x) is the decreasing function that satisfies f for 0 < a < b. When a is an integer and b = a + 1 we deduce several combinatorial results. These include an asymptotic...
Characterization of stationary measures for one-dimensional exclusion processes (2002)
Bramson, Maury, Liggett, Thomas M., Mountford, Thomas
The product Bernoulli measures $\nu_\alpha$ with densities $\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant stationary measures for an exclusion process on $\mathbb{Z}$ with...
Asymptotics Of A Matrix Valued Markov Chain (2002)
Arising In Sociology, Phillip Bonacich, Thomas M. Liggett
We consider a discrete time Markov chain whose state space is the set of all N N stochastic matrices with zero diagonal entries. This chain models the evolution of relationships among N individuals...
How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields (2001)
Holroyd, Alexander E., Liggett, Thomas M.
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...
Monotonicity of conditional distributions and growth models on trees (2000)
We consider a sequence of probability measures $\nu_n$ obtained by conditioning a random vector $X =(X_1,\ldots,X_d)$ with nonnegative integer valued components on $$ X_1 + \dots + X_d = n - 1 $$ and...
Tagged Particle Distributions Or How To Choose A Head At Random (2000)
. Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed by Z d , there exists an occupied site X 2 Z d with the property that...
Izhak Rubin, Thomas M. Liggett
OF THE THESIS Stability and Scheduling in Multiclass Queueing Networks by Cedric Westphal Doctorate of Philosophy in Electrical Engineering University of California, Los Angeles, 1999 Professor...
, find a random occupied site (2000)
Er E. Holroyd, Thomas M. Liggett
such that relative to X, the other random variables are still i.i.d. Bernoulli. Results of Thorisson [9] imply that such an X exists for all d. Liggett [7] proved that for d = 1, there exists an X...
Monotonicity Of Conditional Distributions And Growth Models On Trees (1999)
. We consider a sequence of probability measures n obtained by conditioning a random vector X = (X 1 ; :::; X d ) with nonnegative integer valued components on X 1 + \Delta \Delta \Delta + X d = n...
Stochastic models of interacting systems (1997)
Interacting particle systems is by now a mature area of probability theory, but one that is still very active. We begin this paper by explaining how models from this area arise in fields such as...
Multiple transition points for the contact process on the binary tree (1996)
The contact process on $Z^d$ is known to have only two fundamental types of behavior: survival and extinction. Recently Pemantle discovered that the phase structure for the contact process on a tree...