arXiv subjects

Fernando P. A. Prado

Publications and source records attributed to Fernando P. A. Prado.

5 recordsLinked to original sources

arXiv2026

Reinforced random walks with geometric inter-transition times

We consider interacting vertex-reinforced random walks on a finite graph, each transitioning according to independent geometric holding times of parameter $p_i \in (0,1]$. Letting $x=X(n)$ be the vector of vertex-occupation proportions up to time $n$, the one-step transition probabilities of walk $i$ are governed by $Q^i(x,p_i)=p_iΠ^i(x)+(1-p_i)I$, where $Π^i(x)$ has rows equal to a probability measure $π^i(x)$ on the vertex set and $I$ is the identity. Its unique invariant measure is thus $π^i(x)$, independent of $p_i$. Consequently, the limiting points of $X(n)$ coincide with those of the simultaneous-transition model ($p_i=1$): the solutions of $x=π(x)$. However, almost sure convergence is non-trivial: the standard stochastic-approximation approach requires the Clark-Kushner condition, which is not immediate since the stochastic input is biased by the walk current state. We overcome this via a decomposition of the input into a martingale and a geometrically decaying correction, establishing almost sure convergence.

math.PR↗

arXiv2026

Interacting vertex reinforced random walks on complete sub-graphs

This article introduces a model for interacting vertex-reinforced random walks, each taking values on a complete subgraph of a locally finite undirected graph. The transition probability for a walk to a given vertex depends on the cumulative proportion of visits by all walks that have access to that vertex. Proportions are modified by multiplication by a real valued interaction parameter and the addition of a parameter representing the intrinsic preference of the walk for the vertex. This model covers a wide range of interactions, including the cooperation (attraction) or competition (repulsion) of several walks at single vertices. We are principally concerned with strong laws for the proportion of visits to each vertex by all walks. We prove that this measure converges almost surely towards the set of fixed points of the transition probabilities. Almost sure convergence to a single fixed point is in fact the generic behaviour as we show this to hold for almost all parameter values of our model. Beyond almost sure convergence, our model provides a general framework that yields a detailed description of the limiting behaviour for any choice of interaction parameters and subgraph geometry. We illustrate this by analyzing interacting walks on complete graphs, stars, and cycles, chosen to highlight the model's broad applicability. The central contribution lies in offering a powerful tool to analyse diverse types of interactions mediated by the intersections of subgraphs. Importantly, our results provide not only convergence criteria, but also conditions under which the empirical proportions of the walks' visits fail to converge to certain boundary points of the space where their trajectory evolves.

math.PR↗

arXiv2026

The Clark-Kushner condition for interacting reinforced random walks on finite graphs

We establish the Clark-Kushner condition for a large class of interacting vertex-reinforced random walks on finite graphs, where the transition matrix $Q^i(x)$ of each walk depends on the joint vector $x$ of vertex occupation proportions and may have distinct rows. This allows one to study the dynamics of the vertex occupation measure by using the tools of stochastic approximation theory. However, the standard approach fails because the noise inputs are in our case not a martingale difference: they retain memory of the previous state. Using the solution of the Poisson equation for Markov chains, we decompose the noise into a martingale difference minus the increment of a bounded process -- a structure originating in Gordin's work on limit theorems for stationary processes. The key technical ingredient of our approach is a uniform geometric ergodicity bound derived from the Dobrushin contraction coefficient, which also controls the Lipschitz continuity of the solution of the Poisson equation. Our hypotheses require only that each $Q^i(x)$ be irreducible, aperiodic, and Lipschitz continuous in $x$; in particular, strictly positive entries are not assumed. Our results generalize and simplify previous arguments considered for single self-reinforced vertex-reinforced random walks.

math.PR↗

arXiv2021

Two repelling random walks on $\mathbb Z$

We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter $β\geq 0$. When $β= 0$ both processes are independent symmetric random walks on $\mathbb{Z}$, and hence recurrent. We show that both random walks are further recurrent if $β\in (0,1]$. We also show that these processes are transient and diverge in opposite directions if $β> 2$. The case $β\in (1,2]$ remains widely open. Our results are obtained by considering the dynamical system approach to stochastic approximations.

math.PR↗

arXiv2020

Vertex reinforced random walks with exponential interaction on complete graphs

We describe a model for $m$ vertex reinforced interacting random walks on complete graphs with $d\geq 2$ vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if the absolute value of the interaction strength parameters are smaller than a certain upper bound, then, for any number of random walks ($m\geq 2$) on any graph ($d \geq 2$), the vertex occupation measure converges toward a unique equilibrium. We provide two additional examples of repelling random walks for the cases $m=d=2$ and $m=3$, $d=2$. The latter is used to study some properties of three exponentially repelling random walks on $\mathbb{Z}$.

math.PR↗