arXiv subjects

Luca Avena

Publications and source records attributed to Luca Avena.

32 records · Page 2Linked to original sources

arXiv2020

Loop-erased partitioning of a graph: mean-field analysis

We consider a random partition of the vertex set of an arbitrary graph that can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$, that we see as a tuning parameter.The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block of the random partition. This interaction potential can be seen as an affinity measure for ``densely connected nodes'' and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structures. For the latter geometry we show a phase-transition for ``community detectability'' as a function of the tuning parameter and the edge weights.

math.PR↗

arXiv2019

Random walk in cooling random environment: recurrence versus transience and mixed fluctuations

This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. We show that the answer depends in a delicate way on the choice of the refreshing times. An overarching goal of our paper is to investigate how the behaviour of a random process with a rich correlation structure can be affected by resettings.

math.PR↗

arXiv2019

Estimating the inverse trace using random forests on graphs

Some data analysis problems require the computation of (regularised) inverse traces, i.e. quantities of the form $\Tr (q \bI + \bL)^{-1}$. For large matrices, direct methods are unfeasible and one must resort to approximations, for example using a conjugate gradient solver combined with Girard's trace estimator (also known as Hutchinson's trace estimator). Here we describe an unbiased estimator of the regularized inverse trace, based on Wilson's algorithm, an algorithm that was initially designed to draw uniform spanning trees in graphs. Our method is fast, easy to implement, and scales to very large matrices. Its main drawback is that it is limited to diagonally dominant matrices $\bL$.

stat.CO↗

arXiv2018

Intertwining wavelets or Multiresolution analysis on graphs through random forests

We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q'. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q'. We illustrate the method by numerical experiments.

cs.IT↗

arXiv2018

Random walks on dynamic configuration models: a trichotomy

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction $α_n$ of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as $n\to\infty$, when $α_n$ is chosen such that $\lim_{n\to\infty} α_n (\log n)^2 = β\in [0,\infty]$. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order $1/\sqrt{α_n}$ when $β=\infty$. In the present paper we investigate what happens when $β\in [0,\infty)$. It turns out that the mixing time is of order $\log n$, with the scaled mixing time exhibiting a one-sided cutoff when $β\in (0,\infty)$ and a two-sided cutoff when $β=0$. The occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes from a competition between the time scales of mixing on the static graph, as identified by Ben-Hamou and Salez [4], and the regeneration time of first stepping across a rewired edge.

math.PR↗

arXiv2018

Mixing times of random walks on dynamic configuration models

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random graph becomes {\em dynamic}, namely, at each unit of time a fraction $α_n$ of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every $\varepsilon\in(0,1)$ the $\varepsilon$-mixing time of random walk without backtracking grows like $\sqrt{2\log(1/\varepsilon)/\log(1/(1-α_n))}$ as $n \to \infty$, provided that $\lim_{n\to\infty} α_n(\log n)^2=\infty$. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.

math.PR↗

arXiv2017

Approximate and exact solutions of intertwining equations through random spanning forests

For different reversible Markov kernels on finite state spaces, we look for families of probability measures for which the time evolution almost remains in their convex hull. Motivated by signal processing problems and metastability studies we are interested in the case when the size of such families is smaller than the size of the state space, and we want such distributions to be with small overlap among them. To this aim we introduce a squeezing function to measure the common overlap of such families, and we use random forests to build random approximate solutions of the associated intertwining equations for which we can bound from above the expected values of both squeezing and total variation errors. We also explain how to modify some of these approximate solutions into exact solutions by using those eigenvalues of the associated Laplacian with the largest absolute values.

math.PR↗

arXiv2016

The parabolic Anderson model on the hypercube

We consider the parabolic Anderson model $\frac{\partial}{\partial t} v_n=κΔ_n v_n + ξ_n v_n$ on the $n$-dimensional hypercube $\{-1,+1\}^n$ with random i.i.d. potential $ξ_n$. We parametrize time by volume and study $v_n$ at the location of the $k$-th largest potential, $x_{k,2^n}$. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales $v_n(t_n,x_{k,2^n})$ behaves like a system without diffusion and grows as $\exp\big\{(ξ_n(x_{k,2^n}) - κ)t_n\big\}$, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the operator $κΔ_n+ξ_n$, for which we give precise asymptotics. Moreover, the transition time depends only on the difference $ξ_n(x_{1,2^n})-ξ_n(x_{k,2^n})$. One of our main motivations in this article is to investigate the mutation-selection model of population genetics on a random fitness landscape, which is given by the ratio of $v_n$ to its total mass, with $ξ_n$ corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at $x_{k,2^n}$ moves completely to $x_{1,2^n}$ on time scales where the transition of growth rates happens. The class of potentials we consider involves the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.

math.PR↗

arXiv2016

A class of random walks in reversible dynamic environment: antisymmetry and applications to the East model

We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in http://arxiv.org/abs/1602.06322. As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical interfaces between empty and occupied regions. We study the asymptotic velocity and density profile for the environment seen from the walker. In particular, we determine the sign of the velocity when the density of the underlying East process is not 1/2, and we discuss the appearance of a drift in the balanced setting given by density 1/2.

math.PR↗

arXiv2016

Explicit LDP for a slowed RW driven by a symmetric exclusion process

We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. Such components have different structures (Gaussian and Poissoinian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function corresponding to the RW is explicit.

math.PR↗

arXiv2016

Random spanning forests, Markov matrix spectra and well distributed points

This paper is a variation on the uniform spanning tree theme. We use random spanning forests to solve the following problem: for a Markov process on a finite set of size $n$, find a probability law on the subsets of any given size $m \leq n$ with the property that the mean hitting time of such a random target does not depend on the starting point of the random walk. We then explore the connection between random spanning forests and infinitesimal generator spectrum. In particular we give an almost probabilistic proof of an algebraic result due to Micchelli and Willoughby and used by Fill and Miclo to study the convergence to equilibrium of reversible Markov chains. We finally introduce some related fragmentation and coalescence processes, emphasizing algorithmic aspects, and give an extension of Burton and Pemantle transfer current theorem to the non reversible case.

math.PR↗

arXiv2014

A local CLT for convolution equations with an application to weakly self-avoiding random walks

We prove error bounds in a central limit theorem for solutions of certain convolution equations. The main motivation for investigating these equations stems from applications to lace expansions, in particular to weakly self-avoiding random walks in high dimensions. As an application we treat such self-avoiding walks in continuous space. The bounds obtained are sharper than the ones obtained by other methods.

math.PR↗

arXiv2014

Symmetric exclusion as a random environment: hydrodynamic limits

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.

math.PR↗

arXiv2012

Transient random walk in symmetric exclusion: limit theorems and an Einstein relation

We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We construct a renewal structure from which a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure follow. We further prove an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.

math.PR↗