arXiv subjects

Justin Salez

Publications and source records attributed to Justin Salez.

At least 19 recordsLinked to original sources

arXiv2026

The local product condition implies cutoff

In the theory of mixing times, a famously wrong conjecture predicts that a sequence of Markov processes exhibits cutoff as soon as the product of their Poincar\'e constant and mixing time diverges. We prove that this statement becomes correct once the Poincar\'e constant $\gamma$ is replaced with its natural non-equilibrium refinement, which we denote by $\gamma_\star$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/\gamma_\star)$. This estimate is sharp, and universal up to standard regularity assumptions: it holds on finite and infinite state spaces and from any initial condition, and it does not require reversibility, nor any kind of a chain rule. In addition, for deterministic initialization we show that $\gamma_\star\ge\kappa$, where $\kappa$ is the Bakry-\'Emery curvature, making our result broadly applicable. Finally, our proof is short and self-contained: we simply follow the classical idea of replacing the total variation distance by the more tractable $\chi^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.

math.PR↗

arXiv2026

Edge-regular graphs with non-negative curvature have polynomial growth

A long-standing conjecture in the emerging discrete Bakry-\'Emery theory asserts that bounded-degree graphs satisfying $\mathrm{CD}(0,\infty)$ have polynomial growth. In the present paper, we prove this conjecture for all edge-regular graphs, and even obtain a volume doubling estimate with a constant that depends only on the degree. This is made possible thanks to the discovery of a surprising self-improvement phenomenon, which seems of independent interest: any edge-regular graph satisfying $\mathrm{CD}(\kappa,\infty)$ for some $\kappa\in\mathbb R$ must in fact satisfy $\mathrm{CD}(\kappa,n)$ for some explicit, universal and optimal dimension parameter $n$.

math.CO↗

arXiv2026

Hyper-contractivity and entropy decay in discrete time

Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,\pi)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with $p< q$ implies a one-step entropy contraction estimate of the form ${\mathrm H}(\mu T\,|\,\pi)\le \theta\, {\mathrm H}(\mu\,|\,\pi)$, with $\theta=p/q$. Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups.

math.PR↗

arXiv2026

How fast does the range of simple random walk grow?

Consider a discrete-time simple random walk $(X_t)_{t\ge 0}$ on an infinite, connected, locally finite simple graph $G$, and let \[ R_t := |\{X_0,\ldots,X_t\}| \] denote its range. The main result of this revised note is that positive vertex isoperimetry already forces linear expected range, with no bounded-degree assumption: if \[ \iota_V(G) := \inf_{0<|S|<\infty} \frac{|\partial_V S|}{|S|} >0, \] then $\E_x R_t \ge c(G)(t+1)$ for every starting vertex $x$ and every $t\ge 0$. The proof is direct: vertex expansion implies an unweighted Dirichlet inequality, which in turn gives a uniform positive escape probability from every vertex. We also record a finite counterpart: in an $n$-vertex finite vertex expander, the expected hitting time of an independent stationary random target is $\Theta(n)$, again with no restriction on degrees. We also record a chain of geometrically growing lollipops for which \[ \E_o R_t \asymp t^{1/3}, \] so the subdiffusive exponent $1/3$ need not be accompanied by superdiffusive oscillations. In particular, for this graph the lower and upper logarithmic exponents of $\E_oR_t$ are both equal to $1/3$. Finally, since Barnes and Feige proved the sharp universal estimate $\E T_n= O(n^3)$ for the $n$-th discovery time, we move our elementary proof of the weaker bound $\E T_n=O(n^3\log n)$ to a later section as a short self-contained argument with a logarithmic loss. We close with a related bounded-degree mixing statement: if the lazy walk has worst-case mixing time $m$, then at least $c\sqrt m$ starting vertices are still noticeably unmixed at time $\lfloor m/2\rfloor$. This final result uses the same commute-time/effective-resistance control of connected sets that appears throughout the paper.

math.PR↗

arXiv2025

A transport approach to the cutoff phenomenon

Substantial progress has recently been made in the understanding of the cutoff phenomenon for Markov processes, using an information-theoretic statistics known as varentropy [Sal23; Sal24; Sal25a; PS25]. In the present paper, we propose an alternative approach which bypasses the use of varentropy and exploits instead a new W-TV transport inequality, combined with a classical parabolic regularization estimate [BGL01; OV01]. While currently restricted to non-negatively curved processes on smooth spaces, our argument no longer requires the chain rule, nor any approximate version thereof. As applications, we recover the main result of [Sal25a] establishing cutoff for the log-concave Langevin dynamics, and extend the conclusion to a widely-used discrete-time sampling algorithm known as the Proximal Sampler.

math.PR↗

arXiv2025

Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. The purpose of these lecture notes is to provide a self-contained introduction to this fascinating question, and to describe its recently-uncovered relations with entropy, curvature and concentration.

math.PR↗

arXiv2025

Intrinsic regularity in the discrete log-Sobolev inequality

The chain rule lies at the heart of the powerful Gamma calculus for Markov diffusions on manifolds, providing remarkable connections between several fundamental notions such as Bakry-\'Emery curvature, entropy decay, and hypercontractivity. For Markov chains on finite state spaces, approximate versions of this chain rule have recently been put forward, with an extra cost that depends on the log-Lipschitz regularity of the considered observable. Motivated by those findings, we here investigate the regularity of extremizers in the discrete log-Sobolev inequality. Specifically, we show that their log-Lipschitz constant is bounded by a universal multiple of $\log d$, where $d$ denotes the inverse of the smallest non-zero transition probability. As a consequence, we deduce that the log-Sobolev constant of any reversible Markov chain on a finite state space is at least a universal multiple of $\kappa/\log d$, where $\kappa$ is the Bakry-\'Emery curvature. This is a sharp discrete analogue of what is perhaps the most emblematic application of the Bakry-\'Emery theory for diffusions. We also obtain a very simple proof of the main result in \cite{MR4620718}, which asserts that the log-Sobolev constant and its modified version agree up to a $\log d$ factor. Our work consolidates the role of the sparsity parameter $\log d$ as a universal cost for transferring results from Markov diffusions to discrete chains.

math.PR↗

arXiv2025

A new cutoff criterion for non-negatively curved chains

The cutoff phenomenon was recently shown to systematically follow from non-negative curvature and the product condition, for all Markov diffusions. The proof crucially relied on a classical \emph{chain rule} satisfied by the carr\'e du champ operator, which is specific to differential generators and hence fails on discrete spaces. In the present paper, we show that an approximate version of this chain rule in fact always holds, with an extra cost that depends on the log-Lipschitz regularity of the considered observable. As a consequence, we derive a new cutoff criterion for non-negatively curved chains on finite spaces. The latter allows us to recover, in a simple and unified way, a number of historical instances of cutoff that had been established through model-specific arguments. Emblematic examples include random walk on the hypercube, random transpositions, random walk on the multislice, or MCMC samplers for popular spin systems such as the Ising and Hard-core models on bounded-degree graphs.

math.PR↗

arXiv2025

Cutoff for non-negatively curved diffusions

We resolve the long-standing problem of elucidating the cutoff phenomenon for a vast and important class of Markov processes, namely Markov diffusions with non-negative Bakry-\'Emery curvature. More precisely, we prove that any sequence of non-negatively curved diffusions exhibits cutoff in total variation as soon as the product condition is satisfied. Our result holds in Euclidean spaces as well as on Riemannian manifolds, and for arbitrary non-random initial conditions. It vastly simplifies, unifies and generalizes a number of isolated works that have established cutoff through a delicate and model-dependent analysis of mixing times. The proof is elementary: we exploit a new simple differential relation between varentropy and entropy to produce a quantitative bound on the width of the mixing window.

math.PR↗

arXiv2024

Concentration of information on discrete groups

Motivated by the Asymptotic Equipartition Property and its recently discovered role in the cutoff phenomenon, we initiate the systematic study of varentropy on discrete groups. Our main result is an approximate tensorization inequality which asserts that the varentropy of any conjugacy-invariant random walk is, up to a universal multiplicative constant, at most that of the free Abelian random walk with the same jump rates. In particular, it is always bounded by the number d of generators, uniformly in time and in the size of the group. This universal estimate is sharp and can be seen as a discrete analogue of a celebrated result of Bobkov and Madiman concerning random d-dimensional vectors with a log-concave density (AOP 2011). A key ingredient in our proof is the fact that conjugacy-invariant random walks have non-negative Bakry-\'Emery curvature, a result which seems new and of independent interest.

math.PR↗

arXiv2024

Entropy factorization via curvature

We develop a new framework for establishing approximate factorization of entropy on arbitrary probability spaces, using a geometric notion known as non-negative sectional curvature. The resulting estimates are equivalent to entropy subadditivity and generalized Brascamp-Lieb inequalities, and provide a sharp modified log-Sobolev inequality for the Gibbs sampler of several particle systems in both continuous and discrete settings. The method allows us to obtain simple proofs of known results, as well as some new inequalities. We illustrate this through various applications, including discrete Gaussian free fields on arbitrary networks, the down-up walk on uniform $n$-sets, the uniform measure over permutations, and the uniform measure on the unit sphere in $\R^n$. Our method also yields a simple, coupling-based proof of the celebrated logarithmic Sobolev inequality for Langevin diffusions in a convex potential, which is one of the most emblematic applications of the Bakry-\'Emery criterion.

math.PR↗

arXiv2024

Entropy and curvature: beyond the Peres-Tetali conjecture

We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step contraction in the Wasserstein distance implies a 1-step contraction in relative entropy, by the same amount. Our result substantially strengthens a recent breakthrough of the second author, and has the advantage of being applicable to arbitrary scales. This leads to a time-varying refinement of the standard Modified Log-Sobolev Inequality (MLSI), which allows us to leverage the well-acknowledged fact that curvature improves at large scales. We illustrate this principle with several applications, including birth and death chains, colored exclusion processes, permutation walks, Gibbs samplers for high-temperature spin systems, and attractive zero-range dynamics. In particular, we prove a MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question.

math.PR↗

arXiv2023

The varentropy criterion is sharp on expanders

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to the maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Despite the accumulation of many examples, this phenomenon is still far from being understood, and identifying the general conditions that trigger it has become one of the biggest challenges in the quantitative analysis of finite Markov chains. Very recently, the author proposed a general sufficient condition for the occurrence of a cutoff, based on a certain information-theoretical statistics called varentropy. In the present paper, we demonstrate the sharpness of this approach by showing that the cutoff phenomenon is actually equivalent to the varentropy criterion for all sparse, fast-mixing chains. Reversibility is not required.

math.PR↗

arXiv2023

Spectral gap and curvature of monotone Markov chains

We prove that the absolute spectral gap of any monotone Markov chain coincides with its optimal Ollivier-Ricci curvature, where the word `optimal' refers to the choice of the underlying metric. Moreover, we provide a new expression in terms of local variations of increasing functions, which has several practical advantages over the traditional variational formulation using the Dirichlet form. As an illustration, we explicitly determine the optimal curvature and spectral gap of the non-conservative exclusion process with heterogeneous reservoir densities on any network, despite the lack of reversibility.

math.PR↗

arXiv2022

Upgrading MLSI to LSI for reversible Markov chains

For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by $\log (1/p)$, where $p$ is the minimum non-zero transition probability. We illustrate this by providing the first log-Sobolev estimate for Zero-Range processes on arbitrary graphs. As another application, we determine the modified log-Sobolev constant of the Lamplighter chain on all bounded-degree graphs, and use it to provide negative answers to two open questions by Montenegro and Tetali (2006) and Hermon and Peres (2018). Our proof builds upon the `regularization trick' recently introduced by the last two authors.

math.PR↗

arXiv2022

Separation cutoff for Activated Random Walks

We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fixed infinite non-amenable graph, with absorption at the boundary and (ii) large finite vertex-transitive graphs with absorption at a single vertex. The latter result settles a conjecture of Levine and Liang. Our proofs rely on the refined analysis of a strong stationary time recently discovered by Levine and Liang and involving the IDLA process.

math.PR↗

arXiv2022

Mixing time and expansion of non-negatively curved Markov chains

We establish three remarkable consequences of non-negative curvature for sparse Markov chains. First, their conductance decreases logarithmically with the number of states. Second, their displacement is at least diffusive until the mixing time. Third, they never exhibit the cutoff phenomenon. The first result provides a nearly sharp quantitative answer to a classical question of Ollivier, Milman and Naor. The second settles a conjecture of Lee and Peres for graphs with non-negative curvature. The third offers a striking counterpoint to the recently established cutoff for non-negatively curved chains with uniform expansion.

math.PR↗

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↗