L. R. G. Fontes — Page 2

arXiv2001

A dynamic one-dimensional interface interacting with a wall

We study a symmetric randomly moving line interacting by exclusion with a wall. We show that the expectation of the position of the line at the origin when it starts attached to the wall satisfies the following bounds: c_1t^{1/4} \le\Eξ_t(0) \le c_2 t^{1/4}\log t The result is obtained by comparison with a ``free'' process, a random line that has the same behavior but does not see the wall. The free process is isomorphic to the symmetric nearest neigbor one-dimensional simple exclusion process. The height at the origin in the interface model corresponds to the integrated flux of particles through the origin in the simple exclusion process. We compute explicitly the asymptotic variance of the flux and show that the probability that this flux exceeds Kt^{1/4}\log t is bounded above by const. t^{2-K}. We have also performed numerical simulations, which indicate \Eξ_t(0)^2 \sim t^{1/2}\log t as t\to\infty.

math.PR↗

arXiv2001

Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension

Let $τ= (τ_i : i \in {\Bbb Z})$ denote i.i.d.~positive random variables with common distribution $F$ and (conditional on $τ$) let $X = (X_t : t\geq0, X_0=0)$, be a continuous-time simple symmetric random walk on ${\Bbb Z}$ with inhomogeneous rates $(τ_i^{-1} : i \in {\Bbb Z})$. When $F$ is in the domain of attraction of a stable law of exponent $α<1$ (so that ${\Bbb E}(τ_i) = \infty$ and X is subdiffusive), we prove that $(X,τ)$, suitably rescaled (in space and time), converges to a natural (singular) diffusion $Z = (Z_t : t\geq0, Z_0=0)$ with a random (discrete) speed measure $ρ$. The convergence is such that the ``amount of localization'', $\E \sum_{i \in {\Bbb Z}} [¶(X_t = i|τ)]^2$ converges as $t \to \infty$ to $\E \sum_{z \in {\Bbb R}} [¶(Z_s = z|ρ)]^2 > 0$, which is independent of $s>0$ because of scaling/self-similarity properties of $(Z,ρ)$. The scaling properties of $(Z,ρ)$ are also closely related to the ``aging'' of $(X,τ)$. Our main technical result is a general convergence criterion for localization and aging functionals of diffusions/walks $Y^{(ε)}$ with (nonrandom) speed measures $μ^{(ε)} \to μ$ (in a sufficiently strong sense).

math.PR↗