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T. Mountford

Publications and source records attributed to T. Mountford.

6 recordsLinked to original sources
arXiv2019

Hydrodynamics in a condensation regime: the disordered asymmetric zero-range process

We study asymmetric zero-range processes on Z with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. For any given environment satisfying suitable averaging properties, we establish a hydrodynamic limit given by a scalar conservation law including the domain above critical density, where the flux is shown to be constant.

math.PR↗
arXiv2018

Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder

We study asymmetric zero-range processes on Z with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. We prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities, and dynamic local loss of mass at supercritical hydrodynamic densities. Our results do not assume starting from local Gibbs states. As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In particular , we relax the weak convexity assumption of [7, 8] for the escape of mass property. 1 MSC 2010 subject classification: 60K35, 82C22.

math.PR↗
arXiv2017

Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

We first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\{(s_1,s_2)\in \mathbb{R}^2: X(s_1,s_2) >0\}$ is equal to $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.

math.PR↗
arXiv2015

Supercritical behavior of asymmetric zero-range process with sitewise disorder

We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for asymmetric nearest neighbour zero range processes with non homogeneous jump rates. The class of environments considered is close to that considered by Andjel, Ferrari, Guiol and Landim, while our class of processes is broader. We also give a simpler proof of a result of Ferrari and Sisko with weaker assumptions.

math.PR↗
arXiv2008

Large deviations for voter model occupation times in two dimensions

We study the decay rate of large deviation probabilities of occupation times, up to time $t$, for the voter model $η\colon\Z^2\times[0,\infty)\ra\{0,1\}$ with simple random walk transition kernel, starting from a Bernoulli product distribution with density $ρ\in(0,1)$. Bramson, Cox and Griffeath (1988) showed that the decay rate order lies in $[\log(t),\log^2(t)]$. In this paper, we establish the true decay rates depending on the level. We show that the decay rates are $\log^2(t)$ when the deviation from $ρ$ is maximal (i.e., $η\equiv 0$ or 1), and $\log(t)$ in all other situations.

math.PR↗