arXiv subjects

Pietro Caputo

Publications and source records attributed to Pietro Caputo.

At least 19 recordsLinked to original sources

arXiv2026

Aldous' spectral gap phenomena in stochastic exchange models

We consider a broad class of exchange dynamics with arbitrary weighted graph or hypergraph update structures, which includes the Kipnis--Marchioro--Presutti (KMP) model and the energies of Kac's walk on the sphere. These are conservative continuous-spin systems whose reversible measures are Dirichlet distributions. We prove that their spectral gap is always attained by a polynomial of degree at most two in the energy variables. Equivalently, through an intertwining with the associated discrete particle systems, the dominant mode is always represented by either a one-particle or a two-particle observable. We interpret this as a manifestation of an Aldous-type spectral gap phenomenon. In particular, the resulting two-particle spectral gap identity settles a recent conjecture of Alon and Puder. We also characterize sharply when one particle suffices and when a genuinely two-particle mode dominates. Under a common rescaling of the Dirichlet parameters, segment-like geometries are precisely those for which the gap is always of one-particle type, mean-field geometries are precisely those for which it is always of two-particle type, and all other geometries exhibit a nontrivial transition. The proof relies on the analysis of the so-called hidden model, a dual representation of the original process. The mechanism underlying the degree-two reduction is remarkably robust and extends to a much broader class of stochastic exchange models and their associated particle systems, including the harmonic process, the immediate exchange model, averaging-type processes, and nonreversible variants. Finally, we analyze boundary-driven versions of the KMP model, in which the bulk exchange dynamics interacts with reservoirs, generally resulting in nonreversible processes. In contrast to the conservative setting, we prove that the spectral gap of a boundary-driven model is always of one-particle type.

math.PR↗

arXiv2026

Nonlinear Exchange Dynamics for Independent Sets

In recent years, nonlinear dynamics derived from kinetic theory have gained attention in the context of sampling configurations of spin systems such as the Ising model. We focus on nonlinear dynamics for the hard-core model, a canonical spin system with hard constraints that specifies a distribution over independent sets in a graph, weighted by their sizes. We explore two distinct types of nonlinear dynamics: the mean-field dynamics, which preserves the density (or average size) of independent sets, and the single-site dynamics, which preserves the marginal vector (i.e., the occupancy probabilities of the vertices). These dynamics are natural stochastic processes for sampling from the hard-core model with a specified density or marginal vector, respectively, both of which are canonical instances of maximum entropy distributions that have been studied in various contexts. In contrast to linear Markov chains, there is a significant lack of a fundamental theoretical framework for nonlinear dynamics. We develop foundational theoretical tools for analyzing nonlinear dynamics within the context of the hard-core model. We establish almost linear convergence of both the mean-field and single-site dynamics at sufficiently low density through novel coupling arguments. We also establish exponential decay of relative entropy for the mean-field dynamics all the way up to the critical density. Additionally, we design new algorithms for sampling from the hard-core distribution with either a specified density or a specified marginal vector. These algorithms are based on a related linear Markov chain, called the particle-system dynamics and inspired by the so-called Kac's program, that approximates the associated nonlinear dynamics. As we demonstrate in the paper, they are comparable in time complexity, but simpler to implement, than traditional approaches based on learning parameter values.

cs.DS↗

arXiv2026

$L^2$-cutoff for the averaging process on random regular graphs

We study the mixing time of the averaging process on a large random $d$-regular graph, $d\ge 3$, and prove an $L^2$-cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree $d=10$: for small degrees, i.e., $d\le 10$, the averaging process mixes as fast as the corresponding random walk on the same graph, whereas for $d> 10$ its $L^2$-mixing is governed by a different, slower mechanism. Our proof relies on a detailed asymptotic analysis of an auxiliary biased birth-and-death chain with a slow bond. We also briefly discuss an analogous phase transition for the $L^1$-mixing.

math.PR↗

arXiv2025

Kac's Program and Relative Entropy Decay for Nonlinear Spin-Exchange Dynamics

We introduce and analyze a nonlinear exchange dynamics for Ising spin systems with arbitrary interactions. The evolution is governed by a quadratic Boltzmann-type equation that conserves the mean magnetization. Collisions are encoded through a spin-exchange kernel chosen so that the dynamics converge to the Ising model with the prescribed interaction and mean magnetization profile determined by the initial state. We prove a general convergence theorem, valid for any interaction and any transport kernel. Moreover, we show that, for sufficiently weak interactions, the system relaxes exponentially fast to equilibrium in relative entropy, with optimal decay rate independent of the initial condition. The proof relies on establishing a strong version of the Kac program from kinetic theory. In particular, we show that the associated Kac particle system satisfies a modified logarithmic Sobolev inequality with constants uniform in the number of particles. This is achieved by adapting the method of stochastic localization to the present conservative setting.

math.PR↗

arXiv2025

A universal cutoff phenomenon for mean-field exchange models

We study a broad class of high-dimensional mean-field exchange models, encompassing both noisy and singular dynamics, along with their dual processes. This includes a generalized version of the averaging process as well as some non-reversible extensions of classical exchange dynamics, such as the flat Kac model. Within a unified framework, we analyze convergence to stationarity from worst-case initial data in Wasserstein distance. Our main result establishes a universal cutoff phenomenon at an explicit mixing time, with a precise window and limiting Gaussian profile. The mixing time and profile are characterized in terms of the logarithm of the size-biased redistribution random variable, thus admitting a natural entropic interpretation.

math.PR↗

arXiv2025

Factorizations of relative entropy using stochastic localization

We derive entropy factorization estimates for spin systems using the stochastic localization approach proposed by Eldan and Chen-Eldan, which, in this context, is equivalent to the renormalization group approach developed independently by Bauerschmidt, Bodineau, and Dagallier. The method provides approximate Shearer-type inequalities for the corresponding Gibbs measure at sufficiently high temperature, without restrictions on the degree of the underlying graph. For Ising systems, these are shown to hold up to the critical tree-uniqueness threshold, including polynomial bounds at the critical point, with optimal $O(\sqrt n)$ constants for the Curie-Weiss model at criticality. In turn, these estimates imply tight mixing time bounds for arbitrary block dynamics or Gibbs samplers, improving over existing results. Moreover, we establish new tensorization statements for the Shearer inequality asserting that if a system consists of weakly interacting but otherwise arbitrary components, each of which satisfies an approximate Shearer inequality, then the whole system also satisfies such an estimate.

math.PR↗

arXiv2024

Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time

This paper considers the speed of convergence (mixing) of a finite Markov kernel $P$ with respect to the Kullback-Leibler divergence (entropy). Given a Markov kernel one defines either a discrete-time Markov chain (with the $n$-step transition kernel given by the matrix power $P^n$) or a continuous-time Markov process (with the time-$t$ transition kernel given by $e^{t(P-\mathrm{Id})}$). The contraction of entropy for $n=1$ or $t=0+$ are characterized by the famous functional inequalities, the strong data processing inequality (SDPI) and the modified log-Sobolev inequality (MLSI), respectively. When $P=KK^*$ is written as the product of a kernel and its adjoint, one could also consider the ``half-step'' contraction, which is the SDPI for $K$, while the ``full-step'' contraction refers to the SDPI for $P$. The work [DMLM03] claimed that these contraction coefficients (half-step, full-step, and continuous-time) are generally within a constant factor of each other. We disprove this and related conjectures by working out a number of different counterexamples. In particular, we construct (a) a continuous-time Markov process that contracts arbitrarily faster than its discrete-time counterpart; and (b) a kernel $P$ such that $P^{m+1}$ contracts arbitrarily better than $P^m$. Hence, our main conclusion is that the four standard inequalities comparing five common notions of entropy and variance contraction are generally not improvable. In the process of analyzing the counterexamples, we survey and sharpen the tools for bounding the contraction coefficients and characterize properties of extremizers of the respective functional inequalities. As our examples range from Bernoulli-Laplace model, random walks on graphs, to birth-death chains, the paper is also intended as a tutorial on computing MLSI, SDPI and other constants for these types of commonly occurring Markov chains.

math.PR↗

arXiv2024

Repeated Block Averages: entropic time and mixing profiles

We consider randomized dynamics over the $n$-simplex, where at each step a random set, or block, of coordinates is evenly averaged. When all blocks have size 2, this reduces to the repeated averages studied in [CDSZ22], a version of the averaging process on a graph [AL12]. We study the convergence to equilibrium of this process as a function of the distribution of the block size, and provide sharp conditions for the emergence of the cutoff phenomenon. Moreover, we characterize the size of the cutoff window and provide an explicit Gaussian cutoff profile. To complete the analysis, we study in detail the simplified case where the block size is not random. We show that the absence of a cutoff is equivalent to having blocks of size $n^{\Omega(1)}$, in which case we provide a convergence in distribution for the total variation distance at any given time, showing that, on the proper time scale, it remains constantly 1 up to an exponentially distributed random time, after which it decays following a Poissonian profile.

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

Relaxation time and topology in 1D $O(N)$ models

We discuss the relaxation time (inverse spectral gap) of the one dimensional $O(N)$ model, for all $N$ and with two types of boundary conditions. We see how its low temperature asymptotic behavior is affected by the topology. The combination of the space dimension, which here is always 1, the boundary condition (free or periodic), and the spin state $S^{N-1}$, determines the existence or absence of non-trivial homotopy classes in some discrete version. Such non-trivial topology reflects in bottlenecks of the dynamics, creating metastable states that the system exits at exponential times; while when only one homotopy class exists the relaxation time depends polynomially on the temperature. We prove in the one dimensional case that, indeed, the relaxation time is a proxy to the model's topological properties via the exponential/polynomial dependence on the temperature.

math.PR↗

arXiv2024

Cutoff phenomenon in nonlinear recombinations

We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial marginals. Our main result reveals a cutoff phenomenon for the total variation distance in both discrete and continuous time. Additionally, we derive the explicit cutoff profiles in the case of monochromatic initial distributions. These profiles are different in the discrete and continuous time settings. The proof leverages a pathwise representation of the solution in terms of a fragmentation process associated to a binary tree. In continuous time, the underlying binary tree is given by a branching random process, thus requiring a more elaborate probabilistic analysis.

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

Characterizing Gibbs states for area-tilted Brownian lines

Gibbsian line ensembles are families of Brownian lines arising in many natural contexts such as the level curves of three dimensional Ising interfaces, the solid-on-solid model, multi-layered polynuclear growth etc. An important example is a class of non-intersecting Brownian lines above a hard wall, which are subject to geometrically growing area tilt potentials, which we call the $λ$-tilted line ensemble, where $λ>1$. The model was introduced by Caputo, Ioffe and Wachtel [CIW] in 2018, as a putative scaling limit of the level lines of entropically repulsed solid-on-solid interfaces. In this article we address the problem of classifying all Gibbs measures for $λ$-tilted line ensembles. A stationary infinite volume Gibbs measure was already constructed by [CIW], and the uniqueness of this translation invariant Gibbs measure was recently established by Caputo and Ganguly. Our main result here is a strong characterization for Gibbs measures of $λ$-tilted line ensembles in terms of a two parameter family. Namely, we show that the extremal Gibbs measures are completely characterized by the behavior of the top line $X^1$ at positive and negative infinity, which must satisfy the parabolic growth $$X^{1}(t)=t^2+L\,|t|\,\mathbf{1}_{t<0}+R\,|t|\,\mathbf{1}_{t>0}+o(|t|)\,,\quad \text{ as } \;|t| \to \infty,$$ where $L,R$ are real parameters, including $-\infty,$ with $L+R<0$. The case $L=R=-\infty$ corresponds to the unique translation invariant Gibbs measure. The result bears some analogy to the Airy wanderers, an integrable model introduced and studied in the context of the Airy line ensemble. A crucial step in our proof, of independent significance, is a complete characterization of the extremal Gibbs states associated to a single area-tilted Brownian excursion, which can be interpreted as non-translation invariant versions of the Ferrari-Spohn diffusion.

math.PR↗

arXiv2023

Nonlinear dynamics for the Ising model

We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann's model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely the Ising model at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times $O(n\log n)$ and $O(\log n)$ respectively, where $n$ is the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erd\H{o}s-R\'enyi random graphs, and non-traditional branching processes augmented by a fragmentation process.

math.PR↗

arXiv2023

Uniqueness, mixing, and optimal tails for Brownian line ensembles with geometric area tilt

We consider non-colliding Brownian lines above a hard wall, which are subject to geometrically growing (given by a parameter $\lambda>1$) area tilts, which we call the $\lambda$-tilted line ensemble (LE). The model was introduced by Caputo, Ioffe, Wachtel [CIW] in 2019 as a putative scaling limit for the level lines of low-temperature 3D Ising interfaces. While the LE has infinitely many lines, the case of the single line, known as the Ferrari-Spohn (FS) diffusion, is one of the canonical interfaces appearing in the Kardar-Parisi-Zhang (KPZ) universality class. In contrast with well studied models with determinantal structure such as the Airy LE constructed by Corwin and Hammond as well as the FS diffusion, the $\lambda$-tilted LE is non-integrable. [CIW] constructed a stationary infinite volume Gibbs measure (the zero boundary LE) as a limit of finite LEs on finite intervals with zero boundary conditions, and obtained control on its fluctuations in terms of first moment estimates. Subsequently, Dembo, Lubetzky, Zeitouni revisited the case of finitely many lines and established an equivalence between the free and the zero boundary LEs. In this article we develop probabilistic arguments to resolve several questions that remained open. We prove that the infinite volume zero boundary LE is mixing and hence ergodic and establish a quantitative decay of correlation. Further, we prove an optimal upper tail estimate for the top line matching that of the FS diffusion. Finally, we prove uniqueness of the Gibbs measure in the sense that any uniformly tight LE (a notion which includes all stationary $\lambda$-tilted LE) must be the zero boundary LE. This immediately implies that the LE with free boundary conditions, as the number of lines and the domain size go to infinity arbitrarily converges to this unique LE.

math.PR↗

arXiv2022

Cutoff for the Averaging process on the hypercube and complete bipartite graphs

We consider the averaging process on a graph, that is the evolution of a mass distribution undergoing repeated averages along the edges of the graph at the arrival times of independent Poisson processes. We establish cutoff phenomena for both the $L^1$ and $L^2$ distance from stationarity when the graph is a discrete hypercube and when the graph is complete bipartite. Some general facts about the averaging process on arbitrary graphs are also discussed.

math.PR↗

arXiv2022

Nonlinear recombinations and generalized random transpositions

We study a nonlinear recombination model from population genetics as a combinatorial version of the Kac-Boltzmann equation from kinetic theory. Following Kac's approach, the nonlinear model is approximated by a mean field linear evolution with a large number of particles. In our setting, the latter takes the form of a generalized random transposition dynamics. Our main results establish a uniform in time propagation of chaos with quantitative bounds, and a tight entropy production estimate for the generalized random transpositions, which holds uniformly in the number of particles. As a byproduct of our analysis we obtain sharp estimates on the speed of convergence to stationarity for the nonlinear equation, both in terms of relative entropy and total variation norm.

math.PR↗

arXiv2022

Entropy inequalities for random walks and permutations

We consider a new functional inequality controlling the rate of relative entropy decay for random walks, the interchange process and more general block-type dynamics for permutations. The inequality lies between the classical logarithmic Sobolev inequality and the modified logarithmic Sobolev inequality, roughly interpolating between the two as the size of the blocks grows. Our results suggest that the new inequality may have some advantages with respect to the latter well known inequalities when multi-particle processes are considered. We prove a strong form of tensorization for independent particles interacting through synchronous updates. Moreover, for block dynamics on permutations we compute the optimal constants in all mean field settings, namely whenever the rate of update of a block depends only on the size of the block. Along the way we establish the independence of the spectral gap on the number of particles for these mean field processes. As an application of our entropy inequalities we prove a new subadditivity estimate for permutations, which implies a sharp upper bound on the permanent of arbitrary matrices with nonnegative entries, thus resolving a well known conjecture.

math.PR↗