Pietro Caputo

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.

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.

We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance is 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 an MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question.

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 Pn) or a continuous-time Markov process (with the time-t transition kernel given by et(P−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 [18] claimed that these contraction coefficients (half-step, full-step, and continuous-time) are generally comparable, that is their ratio is bounded from above and below by two absolute constants. 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 Pm+1 contracts arbitrarily better than Pm. Hence, our main conclusion is that the four standard inequalities comparing five common notions of entropy and variance contraction are generally not improvable (even in the subclass of factorizable Markov chains). 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. For example, we show that while the MLSI extremizers always have full support (unless the MLSI constant equals twice the spectral gap), the SDPI extremizers can have partial support. As our examples range from Bernoulli-Laplace model, random walks on graphs, to birth-death chains, the paper is also intended as a survey on computing MLSI, SDPI and other constants for these types of commonly occurring Markov chains.