分数对数凹与扇区稳定多项式：计数平面匹配及更多问题

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $μ$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mix…

Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids

We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze [ABF94] this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result: For the first, we build upon recent results of Adiprasito, Huh, and Ka…

Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polyn…

Isotropy and Log-Concave Polynomials: Accelerated Sampling and High-Precision Counting of Matroid Bases

We define a notion of isotropy for discrete set distributions. If $μ$ is a distribution over subsets $S$ of a ground set $[n]$, we say that $μ$ is in isotropic position if $P[e \in S]$ is the same for all $e\in [n]$. We design a new approximate sampling algorithm that leverages isotropy for the class of distributions $μ$ that have a log-concave generating polynomial; this class includes determinantal point processes, strongly Rayleigh distributions, and uniform distributions over matroid bases. We show that when $μ$ is in approximately isotropic position, the running time of our algorithm depe…

具有对数凹签名的 Holant 问题的 FPTAS

FPTAS for Holant Problems with Log-Concave Signatures

本文针对度数有界图中的 $b$-matching 计数及具有对数凹签名的 Holant 问题，基于 Moitra 的 LP 方法并提出扩展耦合树，对 Chen–Gu 耦合去随机化，从而给出 FPTAS。

谱独立性的普适性及其在自旋玻璃快速混合中的应用

Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

该研究证明谱独立性具有普适性，即O(n)弛豫时间可推出谱独立性；还证明哈密顿量多线性延拓光滑的分布的Glauber动力学有O(n)弛豫时间，且这类分布分数对数凹、满足修正对数Sobolev不等式，还得到高温混合p-spin模型的最优O~(n)混合时间，对多领域有重要意义。