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Register a new userHydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances
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Consider a system of particles performing nearest neighbor random walks on the lattice $ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $a$--stable law, $0<a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/alpha}L(N)$, the density profile evolves as the solution of the random equation $partial_t rho = mf L_W rho$, where $mf L_W$ is the generalized second-order differential operator $frac d{du} frac d{dW}$ in which $W$ is a double sided $a$--stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array ${xi_{N,x} : xinbb Z}$ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.
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We study variable-speed random walks on $mathbb Z$ driven by a family of nearest-neighbor time-dependent random conductances ${a_t(x,x+1)colon xinmathbb Z, tge0}$ whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze.
We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice. The particles can be created/annihilated at the boundaries with time-dependent rate. These boundary dynamics are properly accelerated. We prove the hydrodynamic limit of the particle density profile, under the hyperbolic space-time rescaling, evolves with the entropy solution to Burgers equation with Dirichlet boundary conditions. The boundary conditions are characterised by boundary entropy flux pair.
The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp{a}lainen and Balazs recently proved this conjecture for the nearest neighbor case. We extend their results to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.
We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of valleys of height $log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.
We consider two-dimensional marked point processes which are Gibbsian with a two-body-potential U. U is supposed to have an internal continuous symmetry. We show that under suitable continuity conditions the considered processes are invariant under the given symmetry. We will achieve this by using Ruelle`s superstability estimates and percolation arguments.
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