Bernoulli Product Measure Research Articles

Zero-Temperature Stochastic Ising Model on Planar Quasi-Transitive Graphs

We study the zero-temperature stochastic Ising model on some connected planar quasi-transitive graphs, which are invariant under rotations and translations. The initial spin configuration is distributed according to a Bernoulli product measure with parameter p∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p\\in (0,1) $$\\end{document}. In particular, we prove that if p=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p=1/2 $$\\end{document} and the graph underlying the model satisfies the planar shrink property then all vertices flip infinitely often almost surely.

Quantitative ergodicity for the symmetric exclusion process with stationary initial data

We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

We consider the weakly asymmetric simple exclusion process on the discrete space {1,⋯,n-1}(n∈ℕ), in contact with stochastic reservoirs, both with density ρ∈(0,1) at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter ρ. Under time diffusive scaling tn 2 and for ρ=1 2, when the asymmetry parameter is taken of order 1/n, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and to exhibit different boundary behavior than the Cole–Hopf solution.

Exact solution of the facilitated totally asymmetric simple exclusion process

We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASEP a particle at site j in jumps, at integer times, to site j + 1, provided site j − 1 is occupied and site j + 1 is empty. When started with a Bernoulli product measure at density , the system approaches a stationary state. This non-equilibrium steady state (NESS) has phase transitions at and . The different density regimes , , and exhibit many surprising properties; for example, the pair correlation satisfies, for all , , with k = 2 when , k = 6 when , and k = 3 when . The quantity , where is the variance in the number of particles in an interval of length L, jumps discontinuously from to 0 when and when .

The invariant measures and the limiting behaviors of the facilitated TASEP

We study the facilitated totally asymmetric exclusion process on the one dimensional integer lattice. A particle jumps to right at rate one provided that the target site is empty and that the left neighboring site of the particle is occupied. We investigate the invariant measures and the limiting behaviors of the process. We mainly prove the non-existence of spatial ergodic non-degenerate invariant measures having particle density less than or equal to 1∕2, and derive the limiting distribution of the process when the initial distribution is the Bernoulli product measure with density less than or equal to 1∕2. We also prove that in the low density regime, the system finally converges to an absorbing state.

Limit theorems for the tagged particle in exclusion processes on regular trees

We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure.

Stochastic domination in space‐time for the contact process

AbstractLiggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d. Bernoulli product measure. In particular, they proved this for and (for infection rate sufficiently large) d‐ary homogeneous trees Td. In this paper, we prove some space‐time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on Td with . We show that, for large infection rate, there exists a subset Δ of the vertices of Td, containing a “positive fraction” of all the vertices of Td, such that the following holds: The contact process on Td observed on Δ stochastically dominates an independent spin‐flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on observed on certain d‐dimensional space‐time slabs stochastically dominates an i.i.d. Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.

Projections of planar Mandelbrot random measures

Let μ be a planar Mandelbrot measure and π⁎μ its orthogonal projection on one of the principal axes. We study the thermodynamic and geometric properties of π⁎μ. We first show that π⁎μ is exact dimensional, with dim⁡(π⁎μ)=min⁡(dim⁡(μ),dim⁡(ν)), where ν is the Bernoulli product measure obtained as the expectation of π⁎μ. We also prove that π⁎μ is absolutely continuous with respect to ν if and only if dim⁡(μ)>dim⁡(ν). Our results provides a new proof of Dekking–Grimmett–Falconer formula for the Hausdorff and box dimension of the topological support of π⁎μ, as well as a new variational interpretation. We obtain the free energy function τπ⁎μ of π⁎μ on a wide subinterval [0,qc) of R+. For q∈[0,1], it is given by a variational formula which sometimes yields phase transitions of order larger than 1. For q>1, it is given by min⁡(τν,τμ), which can exhibit first order phase transitions. This is in contrast with the analyticity of τμ over [0,qc). Also, we prove the validity of the multifractal formalism for π⁎μ at each α∈(τπ⁎μ′(qc−),τπ⁎μ′(0+)].

Crossover to the Stochastic Burgers Equation for the WASEP with a Slow Bond

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^\beta}2}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac1{\vphantom{n^\beta}2}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\frac{\alpha}{2n^\beta}+\frac{a}{2n^{\vphantom{\beta}\gamma}}$ (resp. $\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\vphantom{\beta}\gamma}}$). Above, $\alpha>0$, $\gamma\geq \beta\geq 0$, $a\geq 0$. For $\beta<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $\gamma>\frac12$, while for $\gamma = \frac12$ it is an energy solution of the stochastic Burgers equation. For $\gamma\geq\beta=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $\gamma\geq\beta> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.

Facilitated Oriented Spin Models: Some Non Equilibrium Results

We perform the rigorous analysis of the relaxation to equilibrium for some facilitated or kinetically constrained spin models (KCSM) when the initial distribution ν is different from the reversible one, μ. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. We concentrate on two basic oriented KCSM: the East model on ℤ, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the AD model on the binary tree introduced in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945–975, 2002), for which the constraint requires the two children to be vacant. It is important to observe that, while the former model is ergodic at any p≠1, the latter displays an ergodicity breaking transition at p c =1/2. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if ν is concentrated on any configuration which does not contain a forever blocked site or if ν is a Bernoulli(p′) product measure for any p′≠1. For the model on the binary tree we prove similar results in the regime p,p′<p c and under the (plausible) assumption that the spectral gap is positive for p<p c . By constructing a proper test function, we also prove that if p′>p c and p≤p c convergence to equilibrium cannot occur for all local functions. Finally, in a short appendix, we present a very simple argument, different from the one given in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945–975, 2002), based on a combination of some combinatorial results together with “energy barrier” considerations, which yields the sharp upper bound for the spectral gap of East when p↑1.

Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder

For a sequence of i.i.d. random variables $\{\xi_x : x\in \bb Z\}$ bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $\xi_x$. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{\xi_x : x\in \bb Z\}$. We prove that the position of the tagged particle converges under diffusive scaling to a Gaussian process if the other particles are initially distributed according to a Bernoulli product measure associated to a smooth profile $\rho_0:\bb R\to [0,1]$.

Central limit theorem for a tagged particle in asymmetric simple exclusion

We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the tagged particle at time t depends on the initial configuration, through the number of empty sites in the interval [ 0 , ( p − q ) α t ] divided by α , on the hyperbolic time scale and on a longer time scale, namely N 4 / 3 .

Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion

We prove a nonequilibrium central limit theorem for the position of a tagged particle in the one-dimensional nearest neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile ρ 0 : R → [ 0 , 1 ] .

Regularity of diffusion coefficient for nearest neighbor asymmetric simple exclusion on [formula omitted

We consider the nearest neighbor asymmetric exclusion process on Z , in which particles jump with probability p ( 1 ) to the right and p ( - 1 ) to the left. Let q = p ( 1 ) / p ( - 1 ) and denote by ν q an ergodic component of the reversible Bernoulli product measure which places a particle at x with probability q x / ( 1 + q x ) . It is well known that under some hypotheses on a local function V, ( 1 / t ) ∫ 0 t V ( η s ) d s converges to a normal distribution with variance σ 2 = σ 2 ( q ) , which depends on q. We prove in this article that σ 2 ( q ) is a C ∞ function of q on ( 0 , 1 ) .

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space {0,1} ℤ2 where the initial configuration ω_0 is chosen according to a Bernoulli product measure, 1’s are stable, and 0’s become 1’s if they are surrounded by at least three neighboring 1’s. In this paper we show that the configuration ω_n at time n converges exponentially fast to a final configuration $$\bar\omega$$ , and that the limiting measure corresponding to $$\bar\omega$$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents β, η, ν and γ, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ℤ2 (i.e. for independent *-percolation on ℤ), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents.This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement [Aizenman and Grimmett, J. Stat. Phys. 63: 817--835 (1991); Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999)

Fluctuations in the occupation time of a site in the asymmetric simple exclusion process

We consider the simple asymmetric exclusion process with nonzero drift under the stationary Bernoulli product measure at density $\rho$. We prove that for dimension $d=2$ the occupation time of the site 0 is diffusive as soon as $\rho\neq 1/2$. For dimension $d=1$, if the density $\rho$ is equal to $1/2$, we prove that the time t variance of the occupation time of the site 0 diverges in a certain sense at least as $t^{5/4}$.

Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature

We study a class of continuous time Markov processes, which describes ± 1 spin flip dynamics on the hypercubic latticeℤ d , d≥ 2, with initial spin configurations chosen according to the Bernoulli product measure with density p of spins + 1. During the evolution the spin at each site flips at rate c= 0, or 0 0, q(t) ≤ exp(−t (1/ d ) −ɛ), for large t. In d= 2 we obtain the complementary bound: for arbitrary ɛ > 0, q(t) ≥ exp(−t (1/2) +ɛ), for large t.

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d ≥ 3 under diffu- sive rescaling starting from the Bernoulli product measure with density 0 <α <1. We prove that the density fluctuation field Y N t converges to a generalized Ornstein-Uhlenbeck process, which is formally the solution of the stochastic differential equation dYt = AYt dt + dB ∇ , where A is a second order differential operator and B ∇ t is a mean zero Gaussian field with known covariances.

ABpercolation on plane triangulations is unimodal

Let ℱ be a countable plane triangulation embedded in ℝ2in such a way that no bounded region contains more than finitely many vertices, and letPpbe Bernoulli (p) product measure on the vertex set of ℱ. LetEbe the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that theAB percolation probability function θΑΒ(p) =Pp(E) is non-decreasing inpfor 0 ≦p≦ ½. By symmetry,θAΒ(p) is therefore unimodal on [0, 1]. This result partially verifies a conjecture due to Halley and stands in contrast to the examples of Łuczak and Wierman.

AB percolation on plane triangulations is unimodal

Let ℱ be a countable plane triangulation embedded in ℝ2in such a way that no bounded region contains more than finitely many vertices, and letPpbe Bernoulli (p) product measure on the vertex set of ℱ. LetEbe the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that theAB percolation probability function θΑΒ(p) =Pp(E) is non-decreasing inpfor 0 ≦p≦ ½. By symmetry,θAΒ(p) is therefore unimodal on [0, 1]. This result partially verifies a conjecture due to Halley and stands in contrast to the examples of Łuczak and Wierman.