# Entropy decay in the Swendsen-Wang dynamics

@article{Blanca2020EntropyDI, title={Entropy decay in the Swendsen-Wang dynamics}, author={Antonio Blanca and Pietro Caputo and Daniela Parisi and Alistair Sinclair and Eric Vigoda}, journal={arXiv: Probability}, year={2020} }

We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb Z}^d$. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is non-local, i.e., it changes the entire configuration in one step. We prove that, whenever strong spatial mixing (SSM) holds, the mixing time on any $n$-vertex cube of ${\mathbb Z}^d$ is $O(\log n)$, improving on the previous best known bound of $O(n)$. SSM is…

## 9 Citations

### Entropy decay in the Swendsen–Wang dynamics on ℤd

- MathematicsSTOC
- 2021

It is proved that, whenever strong spatial mixing (SSM) holds, the mixing time on any n-vertex cube in ℤd is O(logn), and this is tight by establishing a matching lower bound, the previous best known bound was O(n).

### The Swendsen-Wang Dynamics on Trees

- MathematicsAPPROX-RANDOM
- 2021

A novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders is introduced, which implies that the relaxation time is O(1) for all boundary conditions in the uniqueness region or when $\beta_1$ exceeds the uniqueness threshold for the Ising model.

### Low-temperature Ising dynamics with random initializations

- MathematicsSTOC
- 2022

Glauber dynamics on spin systems are well known to suffer exponential slowdowns at low temperatures due to the emergence of multiple metastable phases, separated by narrow bottlenecks that are hard…

### Swendsen-Wang dynamics for the ferromagnetic Ising model with external fields

- Mathematics
- 2022

. We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two…

### On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization

- MathematicsSODA
- 2022

It is proved that a contractive coupling of a local Markov chain implies spectral independence of the Gibbs distribution and shown that spectral independence implies factorization of entropy for arbitrary blocks, establishing optimal bounds on the modified log-Sobolev constant of the corresponding block dynamics.

### Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion

- MathematicsSTOC
- 2021

An optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings is proved and the approximate tensorization of entropy can be deduced from entropy factorization into blocks of fixed linear size.

### Random-cluster dynamics on random graphs in tree uniqueness

- MathematicsArXiv
- 2020

The proof relies on a sharp bound on the "shattering time", i.e., the number of steps required to break up any configuration into $O(\log n)$ sized clusters, to establish rapid mixing of the random-cluster Glauber dynamics on random $\Delta$-regular graphs.

### Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

- ArtCommunications in Mathematical Physics
- 2021

We establish rapid mixing of the random-cluster Glauber dynamics on random Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

### LECTURE NOTES ON ENTROPY AND MARKOV CHAINS

- Physics
- 2022

. Notes for lectures at UC Santa Barbara (Summer 2022) and UC Berkeley (Fall 2022). In progress

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