arXiv subjects

Daniel J. Slonim

Publications and source records attributed to Daniel J. Slonim.

6 recordsLinked to original sources

arXiv2025

Random Walks in Random Environments with Rare Anomalies

We study random walks in i.i.d. random environments on $\mathbb{Z}^d$ when there are two basic types of vertices, which we call "blue" and "red". Each color represents a different probability distribution on transition probability vectors. We introduce a method of studying these walks that compares the expected amount of time spent at a specific site on the event that the site is red with the expected amount of time spent there on the event that the site is blue. This method produces explicit bounds on the asymptotic velocity of the walk. We recover an early result of Kalikow, but with new bounds on the velocity. Next, we consider a "rare anomaly" model where the vast majority of sites are blue, and blue sites are uniformly elliptic, with some almost-sure bounds on the quenched drift. We show that if the red sites satisfy a certain uniform ellipticity assumption in two fixed, non-parallel directions, then even if red sites break the almost-sure bounds on the quenched drift, making red sites unlikely enough lets us obtain bounds on the asymptotic velocity of the walk arbitrarily close to the bounds on the quenced drift at blue sites. Significantly, the required proportion $p^*$ of blue sites to do this does not depend on the distribution of red sites, except through the uniform ellipticity assumption in two directions. Our proof is based on a coupling technique, where two walks run in environments that are the same everywhere except at one vertex. They decouple when they hit that vertex, and our proof is driven by bounds on how long it takes to recouple. We then demonstrate the importance of the i.i.d. assumption by providing a counterexample to the statement of the theorem with this assumption removed. We conclude with open questions.

math.PR↗

arXiv2023

A zero-one law for random walks in random environments on $\mathbb{Z}^2$ with bounded jumps

This paper has two main results, which are connected through the fact that the first is a key ingredient in the second. Both are extensions of results concerning directional transience of nearest-neighbor random walks in random environments to allow for bounded jumps. Zerner and Merkl proved a 0-1 law for directional transience for planar random walks in random environments. We extend the result to non-planar i.i.d. random walks in random environments on $\mathbb{Z}^2$ with bounded jumps. Sabot and Tournier characterized directional transience for a given direction for nearest-neighbor random walks in Dirichlet environments on $\mathbb{Z}^d$, $d\geq1$. We extend this characterization to random walks in Dirichlet environments with bounded jumps.

math.PR↗

arXiv2022

On total weight exiting finite, strongly connected sets in shift-invariant weighted directed graphs on $\mathbb{Z}$

For a shift-invariant weighted directed graph with vertex set $\mathbb{Z}$, we examine the minimal weight $κ_0$ exiting a finite, strongly connected set of vertices. Although $κ_0$ is defined as an infimum, it has been shown that the infimum is always attained by an actual set of vertices. We show that for each underlying directed graph (prior to assignment of the weights), there is a formula for $κ_0$ as a minimum of finitely many integer combinations of the edge weights. We find this formula for several different directed graphs. Motivation for this problem comes from random walks in Dirichlet environments (equivalently, directed edge reinforced random walks), where the size of $κ_0$ has been shown to determine the strength of finite traps where the walk can get stuck for a long time.

math.CO↗

arXiv2022

Ballisticity of Random walks in Random Environments on $\mathbb{Z}$ with Bounded Jumps

We characterize ballistic behavior for general i.i.d. random walks in random environments on $\mathbb{Z}$ with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. Note: This paper duplicates results from some versions of the preprint "Random walks in Dirichlet random environments on $\mathbb{Z}$ with bounded jumps." (arxiv: 2104.14950). The present paper is being split off for reasons of length, and the plan is to remove these results from a future version of the previous paper and replace them with a citation of the present preprint.

math.PR↗

arXiv2021

Random Walks in Dirichlet Random Environments on $\mathbb{Z}$ with Bounded Jumps

We examine a class of random walks in random environments on $\mathbb{Z}$ with bounded jumps, a generalization of the classic one-dimensional model. The environments we study have i.i.d. transition probability vectors drawn from Dirichlet distributions. For this model, we characterize recurrence and transience, and in the transient case we characterize ballisticity. For ballisticity, we give two parameters, $κ_0$ and $κ_1$. The parameter $κ_0$ governs finite trapping effects, and $κ_1$ governs repeated traversals of arbitrarily large regions of the graph. We show that the walk is right-transient if and only if $κ_1>0$, and in that case it is ballistic if and only if $\min(κ_0,κ_1)>1$.

math.PR↗

arXiv2020

Decimation and Interleaving Operations in One-Sided Symbolic Dynamics

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show they are closed under composition. We also study a family of $n$-ary interleaving operations, one for each $n \ge 1$. Given subsets $X_0, X_1, ..., X_{n-1}$ of the shift space, the $n$-ary interleaving operator produces a set whose elements combine individual elements ${\bf x}_i$, one from each $X_i$, by interleaving their symbol sequences cyclically in arithmetic progressions $(\bmod\,n)$. We determine algebraic relations between decimation and interleaving operators and the shift operator. We study set-theoretic $n$-fold closure operations $X \mapsto X^{[n]}$, which interleave decimations of $X$ of modulus level $n$. A set is $n$-factorizable if $X=X^{[n]}$. The $n$-fold interleaving operators are closed under composition and are idempotent. To each $X$ we assign the set $\mathcal{N}(X)$ of all values $n \ge 1$ for which $X= X^{[n]}$. We characterize the possible sets $\mathcal{N}(X)$ as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. This class includes all shift-invariant sets. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable $X$, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

math.DS↗