Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

On the number of solutions in random hypergraph 2-colouring

We determine the limiting distribution of the logarithm of the number of satisfying assignments in the random $k$-uniform hypergraph 2-colouring problem in a certain density regime for all $k\ge 3$ . As a direct consequence we obtain that in this regime the random colouring model is contiguous wrt. the planted model, a result that helps simplifying the transfer of statements between these two models.

Metric Construction, Stopping Times and Path Coupling

In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to \emph{stopping time analysis}. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling with stopping times, using a metric which allows us to restrict analysis to standard one-step path coupling. This approach provides insight for the design of non-standard metrics giving improvements in the analysis of specific problems. We give illustrative applications to hypergraph independ…

On the strength of connectedness of a random hypergraph

Bollobás and Thomason (1985) proved that for each $k=k(n) \in [1, n-1]$, with high probability, the random graph process, where edges are added to vertex set $V=[n]$ uniformly at random one after another, is such that the stopping time of having minimal degree $k$ is equal to the stopping time of becoming $k$-(vertex-)connected. We extend this result to the $d$-uniform random hypergraph process, where $k$ and $d$ are fixed. Consequently, for $m=\frac{n}{d}(\ln n +(k-1)\ln \ln n +c)$ and $p=(d-1)! \frac{\ln n + (k-1) \ln \ln n +c}{n^{d-1}}$, the probability that the random hypergraph models $H_…

迈向马尔可夫链蒙特卡洛的去随机化

Towards derandomising Markov chain Monte Carlo

本文提出了一种基于向后耦合的去随机化框架，通过将对数级随机选择实现对马尔可夫链蒙特卡洛算法的去随机化，并成功应用于超图独立集与超图染色的确定性近似计数。