arXiv subjects

Justin Salez

Publications and source records attributed to Justin Salez.

At least 37 records · Page 2Linked to original sources

arXiv2022

Universality of cutoff for exclusion with reservoirs

We consider the reversible exclusion process with reservoirs on arbitrary networks. We characterize the spectral gap, mixing time, and mixing window of the process, in terms of certain simple statistics of the underlying network. Among other consequences, we establish a non-conservative analogue of Aldous's spectral gap conjecture, and we show that cutoff occurs if and only if the product condition is satisfied. We illustrate this by providing explicit cutoffs on discrete lattices of arbitrary dimensions and boundary conditions, which substantially generalize recent one-dimensional results. We also obtain cutoff phenomena in relative entropy, Hilbert norm, separation distance and supremum norm. Our proof exploits negative dependence in a novel, simple way to reduce the understanding of the whole process to that of single-site marginals. We believe that this approach will find other applications.

math.PR↗

arXiv2021

Cutoff for non-negatively curved Markov chains

Discovered in the context of card shuffling by Aldous, Diaconis and Shahshahani, the cutoff phenomenon has since then been established in a variety of Markov chains. However, proving cutoff remains a delicate affair, which requires a detailed knowledge of the chain. Identifying the general mechanisms underlying this phase transition -- without having to pinpoint its precise location -- remains one of the most fundamental open problems in the area of mixing times. In the present paper, we make a step in this direction by establishing cutoff for Markov chains with non-negative curvature, under a suitably refined product condition. The result applies, in particular, to random walks on abelian Cayley expanders satisfying a mild degree condition, hence in particular to \emph{almost all} abelian Cayley graphs. Our proof relies on a quantitative \emph{entropic concentration principle}, which we believe to lie behind all cutoff phenomena.

math.PR↗

arXiv2021

Sparse expanders have negative curvature

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infinity". We then transfer this result to finite graphs via local weak convergence. The same approach also applies to the Bacry-Emery curvature condition CD$(0,\infty)$, thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).

math.PR↗

arXiv2021

The interchange process on high-dimensional products

We resolve a long-standing conjecture of Wilson (2004), reiterated by Oliveira (2016), asserting that the mixing-time of the unit-rate Interchange Process on the $n$-dimensional hypercube is of order $n$. This follows from a sharp inequality established at the level of Dirichlet forms, from which we also deduce that macroscopic cycles emerge in constant time, and that the log-Sobolev constant of the exclusion process is of order $1$. Beyond the hypercube, our results apply to cartesian products of arbitrary graphs of fixed size, shedding light on a broad conjecture of Oliveira (2013).

math.PR↗

arXiv2020

A sharp log-Sobolev inequality for the multislice

We determine the log-Sobolev constant of the multi-urn Bernoulli-Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant. Our result extends a classical estimate of Lee and Yau (1998) and confirms a conjecture of Filmus, O'Donnell and Wu (2018). Among other applications, we completely quantify the "small-set expansion" phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.

math.PR↗

arXiv2020

Modified log-Sobolev inequalities for strong-Rayleigh measures

We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $π$ satisfies a form of negative dependence known as the stochastic covering property. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where $π$ is $k-$homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle & Peres (2014). As another application, we deduce that the natural Monte-Carlo Markov Chain used to sample from $π$ has mixing time at most $kn\log\log\frac{1}{π(x)}$ when initialized in state $x$. To the best of our knowledge, this is the first work relating negative dependence and modified log-Sobolev inequalities.

math.PR↗

arXiv2019

Entropy dissipation estimates for inhomogeneous zero-range processes

Introduced by Lu and Yau (CMP, 1993), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However, the intractability of certain covariance terms has so far precluded applications to heterogeneous models. Here we demonstrate that the existence of an appropriate coupling can be exploited to bypass this limitation effortlessly. Our main result is a dimension-free modified log-Sobolev inequality for zero-range processes on the complete graph, under the only requirement that all rate increments lie in a compact subset of $(0,\infty)$. This settles an open problem raised by Caputo and Posta (PTRF, 2007) and reiterated by Caputo, Dai Pra and Posta (AIHP, 2009). We believe that our approach is simple enough to be applicable to many systems.

math.PR↗

arXiv2018

A version of Aldous' spectral-gap conjecture for the zero range process

We show that the spectral-gap of a general zero range process can be controlled in terms of the spectral-gap of a single particle. This is in the spirit of Aldous' famous spectral-gap conjecture for the interchange process. Our main inequality decouples the role of the geometry (defined by the jump matrix) from that of the kinetics (specified by the exit rates). Among other consequences, the various spectral-gap estimates that were so far only available on the complete graph or the $d$-dimensional torus now extend effortlessly to arbitrary geometries. As an illustration, we determine the exact order of magnitude of the spectral-gap of the rate-one $\textbf{ZRP}$ on any regular graph and, more generally, for any doubly stochastic jump matrix.

math.PR↗

arXiv2018

Cutoff for the mean-field zero-range process with bounded monotone rates

We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing time from any initial configuration, and establish the cutoff phenomenon. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase: as time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof uses the path coupling technique of Bubley and Dyer, and the analysis of a suitable hydrodynamic limit. To the best of our knowledge, even the order of magnitude of the mixing time was unknown, except in the special case of constant rates.

math.PR↗

arXiv2018

Emergence of extended states at zero in the spectrum of sparse random graphs

We confirm the long-standing prediction that $c=e\approx 2.718$ is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős-Renyi random graph with average degree $c$. This is achieved by a detailed second-order analysis of the resolvent $(A-z)^{-1}$ near the singular point $z=0$, where $A$ is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring $c$. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.

math.PR↗

arXiv2018

Cutoff for the mean-field zero-range process

We study the mixing time of the unit-rate zero-range process on the complete graph, in the regime where the number $n$ of sites tends to infinity while the density of particles per site stabilizes to some limit $ρ>0$. We prove that the worst-case total-variation distance to equilibrium drops abruptly from $1$ to $0$ at time $n\left(ρ+\frac{1}{2}ρ^2\right)$. More generally, we determine the mixing time from an arbitrary initial configuration. The answer turns out to depend on the largest initial heights in a remarkably explicit way. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase. As time passes, the solid phase {dissolves} into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof combines meta-stability, separation of timescale, fluid limits, propagation of chaos, entropy, and a spectral estimate by Morris (2006).

math.PR↗

arXiv2018

Random walk on sparse random digraphs

A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure.

math.PR↗

arXiv2018

Cutoff at the "entropic time" for sparse Markov chains

We study convergence to equilibrium for a large class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix $P$ the mass is essentially concentrated on few entries. Moreover, the random environment is such that rows of $P$ are independent and such that the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium, with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of $P$. As an application, we consider the case where the rows of $P$ are i.i.d. random vectors in the domain of attraction of a Poisson-Dirichlet law with index $α\in(0,1)$. Our main results are based on a detailed analysis of the weight of the trajectory followed by the walker. This approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon.

math.PR↗

arXiv2016

Spectral atoms of unimodular random trees

We use the Mass Transport Principle to analyze the local recursion governing the resolvent $(A-z)^{-1}$ of the adjacency operator of unimodular random trees. In the limit where the complex parameter $z$ approaches a given location $λ$ on the real axis, we show that this recursion induces a decomposition of the tree into finite blocks whose geometry directly determines the spectral mass at $λ$. We then exploit this correspondence to obtain precise information on the pure-point support of the spectrum, in terms of expansion properties of the tree. In particular, we deduce that the pure-point support of the spectrum of any unimodular random tree with minimum degree $δ\ge 3$ and maximum degree $Δ$ is restricted to finitely many points, namely the eigenvalues of trees of size less than $\frac{Δ-2}{δ-2}$. More generally, we show that the restriction $δ\ge 3$ can be weakened to $δ\ge 2$, as long as the anchored isoperimetric constant of the tree remains bounded away from $0$. This applies in particular to any unimodular Galton-Watson tree without leaves, allowing us to settle a conjecture of Bordenave, Sen and Virág (2013).

math.PR↗

arXiv2016

Weighted sampling without replacement

Comparing concentration properties of uniform sampling with and without replacement has a long history which can be traced back to the pioneer work of Hoeffding (1963). The goal of this short note is to extend this comparison to the case of non-uniform weights, using a coupling between the two samples. When the items' weights are arranged in the same order as their values, we show that the induced coupling for the cumulative values is a submartingale coupling. As a consequence, the powerful Chernoff-type upper-tail estimates known for sampling with replacement automatically transfer to the case of sampling without replacement. For general weights, we use the same coupling to establish a sub-Gaussian concentration inequality. We also construct another martingale coupling which allows us to answer a question raised by Luh and Pippenger (2014) on sampling in Polya urns with different replacement numbers.

math.PR↗

arXiv2016

The densest subgraph problem in sparse random graphs

We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erdős-Rényi model, where it settles a conjecture of Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in extending the notion of balanced loads from finite graphs to their local weak limits, using unimodularity. This is a new illustration of the objective method described by Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].

math.PR↗

arXiv2015

Cooperative Strings and Glassy Interfaces

We introduce a minimal theory of glass formation based on the ideas of molecular crowding and resultant string-like cooperative rearrangement, and address the effects of free interfaces. In the bulk case, we obtain a scaling expression for the number of particles taking part in cooperative strings, and we recover the Adam-Gibbs description of glassy dynamics. Then, by including thermal dilatation, the Vogel-Fulcher-Tammann relation is derived. Moreover, the random and string-like characters of the cooperative rearrangement allow us to predict a temperature-dependent expression for the cooperative length $ξ$ of bulk relaxation. Finally, we explore the influence of sample boundaries when the system size becomes comparable to $ξ$. The theory is in agreement with measurements of the glass-transition temperature of thin polymer films, and allows to quantify the temperature-dependent thickness $h_{\textrm{m}}$ of the interfacial mobile layer.

cond-mat.dis-nn↗

arXiv2015

Cutoff for non-backtracking random walks on sparse random graphs

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape.

math.PR↗