arXiv subjects

Zsuzsanna Baran

Publications and source records attributed to Zsuzsanna Baran.

3 recordsLinked to original sources

arXiv2026

Random walk on the small-world network model in 3 or more dimensions

We study the mixing time of a simple random walk on the small-world network defined by adding edges to $\mathbb{Z}_n^d$ as follows: for each pair $\{x,y\}$ we add an edge with probability $Z_n/\|x-y\|^{d}$ with $Z_n$ chosen so that the average number of added edges to every vertex is $1$. When $d\geq 3$, we show that with high probability the mixing time is of order~$\log n$ and that the random walk does not exhibit cutoff.

math.PR↗

arXiv2025

Intersections of branching random walks on $\mathbb{Z}^8$

We consider random walks on $\Z^8$ indexed by the infinite invariant tree, which consists of an infinite spine and finite random trees attached to it on both sides. We establish the precise order of the non-intersection probability between one walk indexed by one side of the tree, and an independent one indexed by both sides of an independent tree. This is analogous to the result by Lawler from the '90s for two independent simple random walks on $\Z^4$. We also prove a weak law of large numbers for the branching capacity of the range of a branching random walk.

math.PR↗

arXiv2023

Phase transition for random walks on graphs with added weighted random matching

For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider whether for a sequence $(G_n)$ of graphs with bounded degrees and corresponding weights $(\varepsilon_n)$, the (weighted) random walk on $(G_n^*)$ has cutoff. For graphs with polynomial growth we show that $\log\left(\frac{1}{\varepsilon_n}\right)\ll\log|V_n|$ is a sufficient condition for cutoff. Under the additional assumption of vertex-transitivity we establish that this condition is also necessary. For graphs where the entropy of the simple random walk grows linearly up to some time of order $\log|V_n|$ we show that $\frac{1}{\varepsilon_n}\ll\log|V_n|$ is sufficient for cutoff. In case of expander graphs we also provide a complete picture for the complementary regime $\frac{1}{\varepsilon_n}\gtrsim\log|V_n|$.

math.PR↗