arXiv subjects

Kuikui Liu

Publications and source records attributed to Kuikui Liu.

21 records · Page 2Linked to original sources

arXiv2020

Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

We say a probability distribution $μ$ is spectrally independent if an associated correlation matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if $μ$ is spectrally independent, then the corresponding high dimensional simplicial complex is a local spectral expander. Using a line of recent works on mixing time of high dimensional walks on simplicial complexes \cite{KM17,DK17,KO18,AL19}, this implies that the corresponding Glauber dynamics mixes rapidly and generates (approximate) samples from $μ$. As an application, we show that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold. This improves the quasi-polynomial running time of Weitz's deterministic correlation decay algorithm \cite{Wei06} for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics \cite{LV97,LV99,DG00,Vig01,EHSVY16}.

cs.DS↗

arXiv2019

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 polynomials. Specifically, if $X$ is a pure simplicial complex with positive weights on its maximal faces, we can associate with $X$ a multiaffine homogeneous polynomial $p_{X}$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_{X}$.

cs.DS↗

arXiv2018

Log-Concave Polynomials III: Mason's Ultra-Log-Concavity Conjecture for Independent Sets of Matroids

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

math.CO↗