arXiv subjects

Guilherme Reis

Publications and source records attributed to Guilherme Reis.

8 recordsLinked to original sources

arXiv2026

Large deviations for marked sparse random graphs with applications to interacting diffusions

We consider the empirical neighborhood distribution of marked sparse Erdős-Rényi random graphs, obtained by decorating edges and vertices of a sparse Erdős-Rényi random graph with i.i.d. random elements taking values on Polish spaces. We prove that the empirical neighborhood distribution of this model satisfies a large deviation principle in the framework of local weak convergence. We rely on the concept of BC-entropy introduced by Delgosha and Anantharam~(2019) which is inspired on the previous work by Bordenave and Caputo~(2015). Our main technical contribution is an approximation result that allows one to pass from graph with marks in discrete spaces to marks in general Polish spaces. As an application of the results developed here, we prove a large deviation principle for interacting diffusions driven by gradient evolution and defined on top of sparse Erdős-Rényi random graphs. In particular, our results apply for the stochastic Kuramoto model. We obtain analogous results for the sparse uniform random graph with given number of edges.

math.PR↗

arXiv2026

Marked random graphs with given degree sequence: large deviations on the local topology

We investigate the behavior of the empirical neighborhood distribution of marked graphs in the framework of local weak convergence. Here we extend known results by considering uniform random graphs with given degree sequences and i.i.d. marks on half-edges and vertices. We establish a large deviation principle for such families of empirical measures. The proof builds on Bordenave and Caputo's seminal 2015 paper, and Delgosha and Anantharam's 2019 introduction of BC entropy, relying on combinatorial lemmas that allow one to construct suitable approximations of measures supported on marked trees. Possible applications of these results are in the study of interacting diffusions on top of random graphs.

math.PR↗

arXiv2023

Interacting Edge-Reinforced Random Walks

We consider the edge-reinforced random walk with multiple (but finitely many) walkers which influence the edge weights together. The walker which moves at a given time step is chosen uniformly at random, or according to a fixed order. First, we consider 2 walkers with linear reinforcement on a line graph comprising three nodes. We show that the edge weights evolve similarly to the setting with a single walker which corresponds to a Pólya urn. In particular, the left edge weight proportion is a martingale at certain stopping times, showing that a (random) limiting proportion exists. We then look at an arbitrary number of walkers on Z with very general reinforcement. We show that in this case, the behaviour is also the same as for a single walker: either all walkers are recurrent or all walkers have finite range. In the particular case of reinforcements of "sequence type", we give a criterion for recurrence.

math.PR↗

arXiv2022

Stochastic processes with competing reinforcements

We introduce a simple but powerful technique to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to non conservative zero range processes on finite graphs, and to multi-particle random walks with positive and negative reinforcement on the edges. The results hold for a broad class of reinforcement functions, including those with superlinear growth. Our technique consists in a comparison of the original processes with suitable reference models. To implement the comparison we estimate a Radon-Nikodym derivative on a carefully chosen set of trajectories. Our results describe the almost surely long time behaviour of the processes. We also prove a phase transition depending on the strength of the reinforcement functions.

math.PR↗

arXiv2022

A proof of Sanov's Theorem via discretizations

We present an alternative proof of Sanov's theorem for Polish spaces in the weak topology that follows via discretization arguments. We combine the simpler version of Sanov's Theorem for discrete finite spaces and well chosen finite discretizations of the Polish space. The main tool in our proof is an explicit control on the rate of convergence for the approximated measures.

math.PR↗

arXiv2022

Large deviations for interacting diffusions with path-dependent McKean-Vlasov limit

We consider a mean-field system of path-dependent stochastic interacting diffusions in random media over a finite time window. The interaction term is given as a function of the empirical measure and is allowed to be non-linear and path dependent. We prove that the sequence of empirical measures of the full trajectories satisfies a large deviation principle with explicit rate function. The minimizer of the rate function is characterized as the path-dependent McKean-Vlasov diffusion associated to the system. As corollary, we obtain a strong law of large numbers for the sequence of empirical measures. The proof is based on a decoupling technique by associating to the system a convenient family of product measures. To illustrate, we apply our results for the delayed stochastic Kuramoto model and for a SDE version of Galves-Löcherbach model.

math.PR↗

arXiv2020

The Directed Edge Reinforced Random Walk: The Ant Mill Phenomenon

We define here a \textit{directed edge reinforced random walk} on a connected locally finite graph. As the name suggests, this walk keeps track of its past, and gives a bias towards directed edges previously crossed proportional to the exponential of the number of crossings. The model is inspired by the so called \textit{Ant Mill phenomenon}, in which a group of army ants forms a continuously rotating circle until they die of exhaustion. For that reason we refer to the walk defined in this work as the \textit{Ant RW}. Our main result justifies this name. Namely, we will show that on any finite graph which is not a tree, and on $\mathbb Z^d$ with $d\geq 2$, the Ant RW almost surely gets eventually trapped into some directed circuit which will be followed forever. In the case of~$\mathbb Z$ we show that the Ant RW eventually escapes to infinity and satisfies a law of large number with a random limit which we explicitly identify.

math.PR↗

arXiv2018

Interacting diffusions on random graphs with diverging degrees: hydrodynamics and large deviations

We consider systems of mean-field interacting diffusions, where the pairwise interaction structure is described by a sparse (and potentially inhomogeneous) random graph. Examples include the stochastic Kuramoto model with pairwise interactions given by an Erdős-Rényi graph. Our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions. For a broad class of interaction functions, we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit, which is given by a McKean-Vlasov diffusion. Moreover, we also prove matching behavior of the two systems at the level of large deviations. Our results extend classical results of dai Pra and den Hollander and provide the first examples of LDPs for systems with sparse random interactions.

math.PR↗