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Omar Boukhadra

Publications and source records attributed to Omar Boukhadra.

7 recordsLinked to original sources
arXiv2025

Anomalous Heat Kernel for Random Walks in Random Environments of Conductances

We study the trapping phenomenon of random walks in random environments of i.i.d. random conductances on the bonds of the grid $\mathbb{Z}^d$, the so-called random conductance model. Our main results concern the important model with conductances in $[0, 1]$ and a polynomial-tailed law near zero for which we find the correct order of decay of the anomalous heat kernel for $d \ge 5$. In $d = 4$, the behavior is found to be normal. In addition, we look at the symmetrical situation with conductances in $[1, \infty)$ with a polynomial law at infinity, which also shows opposite return probability behaviors.

math.PR↗
arXiv2018

On heat kernel decay for the random conductance model

We study discrete time random walks in an environment of i.i.d. non-negative bounded conductances in $\mathbb{Z}^d$. We are interested in the anomaly of the heat-kernel decay. We improve recent results and techniques.

math.PR↗
arXiv2015

Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance Model

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.

math.PR↗
arXiv2012

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on $\Z^4$) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability $\cmss P_ω^{2n}(0,0)$ after $2n$ steps is at most $C(ω) n^{-2} \log n$, but the best lower bound till now has been $C(ω) n^{-2}$. Here we will show that the $\log n$ term marks a real phenomenon by constructing an environment, for each sequence $λ_n\to\infty$, such that $$ \cmss P_ω^{2n}(0,0)\ge C(ω)\log(n)n^{-2}/λ_n, $$ with $C(ω)>0$ a.s., along a deterministic subsequence of $n$'s. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the $d\ge5$ cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.

math.PR↗
arXiv2010

Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances model

We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $ω_{xy}\in[0,1]$ with a power law with an exponent $γ$ near 0. We are interested in estimating the quenched decay of the return probability $P_ω^{t}(0,0)$, as $t$ tends to $+\infty$. We show that for $γ> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order. As an expected concequence, the same result holds for the discrete-time case.

math.PR↗
arXiv2009

Heat-kernel estimates for random walk among random conductances with heavy tail

We study models of discrete-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $ω_{xy}\in[0,1]$, with polynomial tail near 0 with exponent $γ>0$. We first prove for all $d\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $γ$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $γ$.

math.PR↗