arXiv subjects
Publications and source records attributed to Allan Sly.
Counter to the general notion that the regular tree is the worst case for decay of correlation between sets and nodes, we produce an example of a multi-spin interacting system which has uniqueness on the $d$-regular tree but does not have uniqueness on some infinite $d$-regular graphs.
The reconstruction problem on the tree has been studied in numerous contexts including statistical physics, information theory and computational biology. However, rigorous reconstruction thresholds have only been established in a small number of models. We prove the first exact reconstruction threshold in a non-binary model establishing the Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten-Stigum bound is not tight for the $q$-state Potts model when $q \geq 5$. Moreover, we determine asymptotics for the reconstruction thresholds.
Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $Δ$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $Δ\leq \frac12 k\log k - o(k\log k)$ and reconstruction when $Δ\geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $Δ\leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $Δ\geq k[\log k + \log \log k + 1+o(1)]$.
We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long-range dependence is strong enough, the limit is a Hermite process, while for weaker long-range dependence, the limit is $α$-stable Lévy motion. For the critical value of the long-range dependence parameter, the limit is a sum of a Hermite process and $α$-stable Lévy motion.
For a bivariate Lévy process $(ξ_t,η_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t:=e^{ξ_t}(z+\int_0^t e^{-ξ_{s-}}dη_s), t\ge0, where $z\in\mathbb{R}.$ We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the Lévy process and reveal the effect of the dependence relationship between $ξ$ and $η.$ We also present technical results which explain the structure of the lower bound of the GOU.
Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erdős-Rényi random graph G(n,d/n). While the average degree in G(n,d/n) is d(1-o(1)), it contains many nodes of degree of order $\log n / \log \log n$. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring or the Ising model, the mixing time of Gibbs sampling is at least $n^{1 + Ω(1 / \log \log n)}$. High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. In this work consider sampling q-colorings and show that for every $d < \infty$ there exists $q(d) < \infty$ such that for all $q \geq q(d)$ the mixing time of Gibbs sampling on G(n,d/n) is polynomial in $n$ with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n,d/n) for d > 1 where the number of colors does not depend on n. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures.