arXiv subjects

Jhon Astoquillca

Publications and source records attributed to Jhon Astoquillca.

5 recordsLinked to original sources

arXiv2026

Collision Properties in Discrete and Continuous Time on Bounded-Degree Graphs

We prove that, on every connected bounded-degree graph, the infinite collision property holds in discrete time if and only if it holds in constant-speed continuous time, and the same equivalence holds for the finite collision property. The proof is based on a pointwise comparison between the Green kernels of the synchronous discrete-time pair chain and the asynchronous pair chain obtained by Poissonization. The main estimate exploits the binomial interlacing of the coordinate updates together with a local binomial estimate. We then use Martin capacities and collision zero--one laws to relate this Green-kernel comparison to infinite visits of the diagonal.

math.PR↗

arXiv2026

Collisions of random walks in unimodular random graphs: applications to the random geometric graph and long-range percolation

We study collision properties of simple random walks in unimodular random rooted graphs. This work continues the study initiated in~\cite{HutchcroftPeres2015}: under recurrence and an integrability condition on the root, two independent random walks collide infinitely often a.s. We prove that, under transience and an integrability assumption involving the Green function on the root, two independent random walks collide only finitely often a.s. We apply these results to several random graphs with unbounded degree: the Gilbert graph, the Delaunay graph, the Gabriel graph; and the long-range percolation model. We use these collision properties to characterize stationary measures of the voter model on these graphs.

math.PR↗

arXiv2026

Ergodicity of the voter model with dynamic anti-voter bonds

The voter model with anti-voter bonds is a variant of the classical voter model in which the edges of the underlying graph are assigned signs. At each update, a voter chooses a neighbour according to a transition kernel; interactions across a positive edge follow the usual voter dynamics, so that a site adopts the current opinion of its chosen neighbour, whereas interactions across a negative edge lead to the adoption of the opposite opinion. In this work, we introduce a new variant in which the edge signs evolve dynamically according to dynamical percolation with density parameter $p \in (0,1)$ and speed $\mathsf{v} \in (0,\infty)$, where the two states of the process represent positive and negative edges. This defines a joint spin-bond Markov process. Following Liggett's notion of ergodicity, we prove that this process is ergodic on any simple graph with countably many vertices, with an arbitrary transition kernel of adoption rates and for all choices of the parameters of the edge dynamics.

math.PR↗

arXiv2024

Percolation on the stationary distributions of the voter model with stirring

The voter model with stirring is a variant of the classical voter model on $\mathbb{Z}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf v \ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d \ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{ \mu_{\alpha,\mathsf{v}}: \alpha \in [0,1] \}$, where~$\alpha$ is the density of voters with opinion~1. Sampling a configuration~$\xi$ from~$\mu_{\alpha, \mathsf v}$, we study~$\xi$ as a site percolation model on~$\mathbb{Z}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$\xi$. Letting~$\alpha_c(\mathsf v)$ be the supremum of all the values of~$\alpha$ for which percolation does not occur~$\mu_{\alpha, \mathsf v}$-a.s., we prove that $\alpha_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf v$ large enough, the model exhibits a non-trivial phase transition in~$\alpha$.

math.PR↗

arXiv2024

On the stationary measures of two variants of the voter model

In the voter model, vertices of a graph (interpreted as voters) adopt one out of two opinions (0 and 1), and update their opinions at random times by copying the opinion of a neighbor chosen uniformly at random. This process is dual to a system of coalescing random walks. The duality implies that the set of stationary measures of the voter model on a graph is linked to the dynamics of the collision of random walks on this graph. By exploring the key ideas behind this relationship, we characterize the sets of stationary measures for two variations of the voter model: first, a version that incorporates interchanging of opinions among voters, and second, the voter model on dynamical percolation. To achieve these results, we analyze the collision properties of random walks in two contexts: first, with a swapping behavior that complicates collisions, and second, with random walks defined on a dynamical percolation environment.

math.PR↗