Mental Health and Homelessness

Probability Theory and Related Fields, 2007

We examine the long-range exclusion process introduced by Spitzer and studied by Liggett and answer some of the open questions raised by Liggett. In particular, we show the existence of equilibria corresponding to bounded dual harmonic functions and that the process can have right-discontinuous paths at strictly positive times. We also show that "explosions" when they occur, do so at fixed times determined by the initial configuration. Finally, we give an example for which the configuration with all sites occupied is not stable although the rate at which particles arrive at any given site for that configuration is infinite.

The Annals of Probability, 2005

We show that if µ is an invariant measure for the long range exclusion process putting no mass on the full configuration, L is the formal generator of that process and f is a cylinder function, then Lf ∈ L 1 (dµ) and Lf dµ = 0. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on Z d when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on Z when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.

Hydrodynamic Limits and Related Topics, 2000

We prove a weak law of large numbers for a tagged particle in a totally asymmetric exclusion process on the one-dimensional lattice. The particles are allowed to take long jumps but not pass each other. The object of the paper is to illustrate a special technique for proving such theorems. The method uses a coupling that mimics the Hopf-Lax formula from the theory of viscosity solutions of Hamilton-Jacobi equations.

Springer Proceedings in Mathematics & Statistics, 2014

In this paper we consider exclusion processes {η t : t ≥ 0} evolving on the one-dimensional lattice Z, under the diffusive time scale tn 2 and starting from the invariant state ν ρ-the Bernoulli product measure of parameter ρ ∈ [0, 1]. Our goal consists in establishing the scaling limits of the additive functional Γ t := ∫ tn 2 0 η s (0) ds-the occupation time of the origin. We present a method, recently introduced in [7], from which a local Boltzmann-Gibbs Principle can be derived for a general class of exclusion processes. In this case, this principle says that Γ t is very well approximated to the additive functional of the density of particles. As a consequence, the scaling limits of Γ t follow from the scaling limits of the density of particles. As examples we present the mean-zero exclusion, the symmetric simple exclusion and the weakly asymmetric simple exclusion. For the latter under a strong asymmetry regime, the limit of Γ t is given in terms of the solution of the KPZ equation.

Journal of Statistical Physics, 2004

Stochastic lattice gases with degenerate rates, namely conservative particle systems where the exchange rates vanish for some configurations, have been introduced as simplified models for glassy dynamics. We introduce two particular models and consider them in a finite volume of size ℓ in contact with particle reservoirs at the boundary. We prove that, as for non-degenerate rates, the inverse of the spectral gap and the logarithmic Sobolev constant grow as ℓ 2. It is also shown how one can obtain, via a scaling limit from the logarithmic Sobolev inequality, the exponential decay of a macroscopic entropy associated to a degenerate parabolic differential equation (porous media equation). We analyze finally the tagged particle displacement for the stationary process in infinite volume. In dimension larger than two we prove that, in the diffusive scaling limit, it converges to a Brownian motion with non-degenerate diffusion coefficient.

Physical Review E, 2014

In an exclusion process with avalanches, each particle can hop to a neighboring empty site and if this site is adjacent to an island, the particle on the other end of the island immediately hops and if it joins another island this triggers another hop. There are no restrictions on the length of islands and the duration of the avalanche. This process is well-defined in the low-density region, ρ < 1 2 . We describe the nature of steady states (on a ring) and determine all correlation functions. For the asymmetric version we compute the steady state current, and describe shock and rarefaction waves which arise in the evolution of the step-function initial profile. For the symmetric version we determine the diffusion coefficient and we also examine diffusion of a tagged particle.

2011

The simple exclusion process is formally defined as follows : each particle performs a simple random walk on a set of sites and interacts with other particles by never moving on occupied sites. Despite its simplicity, this process has properties that are found in many more complex statistical mechanics models. It is the combination of the simplicity of the process and the importance of the observed phenomena that make it one of the reference models in out of equilibrium statistical mechanics. In this thesis, I’m interested in the case of the totally asymmetric exclusion process (particles jump only to the right) on N to study its behavior according to the mechanism of particle creation : particles are created at site 0 with arate depending on the current configuration. Once this mechanism is no longer a Poisson process, the associated exclusion process does not admit a product invariant measure. As a consequence, classical computation methods with theinfinitesimal generator are rare...

arXiv (Cornell University), 2020

We describe the translation invariant stationary states of the one dimensional discrete-time facilitated totally asymmetric simple exclusion process (F-TASEP). In this system a particle at site j in Z jumps, at integer times, to site j + 1, provided site j − 1 is occupied and site j + 1 is empty. This defines a deterministic noninvertible dynamical evolution from any specified initial configuration on {0, 1} Z. When started with a Bernoulli product measure at density ρ the system approaches a stationary state, with phase transitions at ρ = 1/2 and ρ = 2/3. We discuss various properties of these states in the different density regimes 0 < ρ < 1/2, 1/2 < ρ < 2/3, and 2/3 < ρ < 1; for example, we show that the pair correlation g(j) = η(i)η(i+ j) satisfies, for all n ∈ Z, k(n+1) j=kn+1 g(j) = kρ 2 , with k = 2 when 0 ≤ ρ ≤ 1/2 and k = 3 when 2/3 ≤ ρ ≤ 1, and conjecture (on the basis of simulations) that the same identity holds with k = 6 when 1/2 ≤ ρ ≤ 2/3. The ρ < 1/2 stationary state referred to above is also the stationary state for the deterministic discrete-time TASEP at density ρ (with Bernoulli initial state) or, after exchange of particles and holes, at density 1 − ρ.

Communications in Mathematical Physics, 2001

We prove that the self-diffusion coefficient of a tagged particle in the symmetric exclusion process in Z d , which is in equilibrium at density α, is of class C ∞ as a function of α in the closed interval [0, 1]. The proof provides also a recursive method to compute the Taylor expansion at the boundaries.

2008