Jonathan Hermon

Bio: Jonathan Hermon is an academic researcher from University of British Columbia. The author has contributed to research in topics: Degree (graph theory) & Order (ring theory). The author has an hindex of 12, co-authored 67 publications receiving 403 citations. Previous affiliations of Jonathan Hermon include University of California, Berkeley & University of São Paulo.

Characterization of cutoff for reversible Markov chains

Abstract: A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least $\alpha$, for some $\alpha\in(0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some “worst” set of stationary measure at least $\alpha$ was not hit by time $t$. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to $\infty$.

Modified log-Sobolev inequalities for strong-Rayleigh measures

Abstract: We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $\pi$ satisfies a form of negative dependence known as the stochastic covering property. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where $\pi$ is $k-$homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle & Peres (2014). As another application, we deduce that the natural Monte-Carlo Markov Chain used to sample from $\pi$ has mixing time at most $kn\log\log\frac{1}{\pi(x)}$ when initialized in state $x$. To the best of our knowledge, this is the first work relating negative dependence and modified log-Sobolev inequalities.

Rapid Mixing of Hypergraph Independent Set

Abstract: We prove that the the mixing time of the Glauber dynamics for sampling independent sets on $n$-vertex $k$-uniform hypergraphs is $O(n\log n)$ when the maximum degree $\Delta$ satisfies $\Delta \leq c 2^{k/2}$, improving on the previous bound [BDK06] of $\Delta \leq k-2$. This result brings the algorithmic bound to within a constant factor of the hardness bound of [BGG+16] which showed that it is NP-hard to approximately count independent sets on hypergraphs when $\Delta \geq 5 \cdot 2^{k/2}$.

Characterization of cutoff for reversible Markov chains

Abstract: A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least α, for some α ∈ (0, 1).We also give general bounds on the total variation distance of a reversible chain at time t in terms of the probability that some "worst" set of stationary measure at least α was not hit by time t. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to ∞.

Frogs on trees

Abstract: We study a system of simple random walks on $\mathcal{T} _{d,n}=({\cal V}_{d,n},{\cal E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda $) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o} $. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o} $. Active particles perform independent simple random walk on the tree of length $ t \in{\mathbb N} \cup \{\infty \} $, referred to as the particles’ lifetime. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\mathcal{R} _t$ be the set of vertices which are visited by the process (with lifetime $t$). The susceptibility ${\mathcal S}({\mathcal T}_{d,n}):=\inf \{t:\mathcal{R} _t={\cal V}_{d,n} \} $ is the minimal lifetime required for the process to visit all sites. The cover time $\mathrm{CT} ({\mathcal T}_{d,n})$ is the first time by which every vertex was visited at least once, when we take $t=\infty $. We show that there exist absolute constants $c,C>0$ such that for all $d \ge 2$ and all $\lambda = {\lambda }_n >0$ which does not diverge nor vanish too rapidly as a function of $n$, with high probability $c \le \lambda{\mathcal S} ({\mathcal T}_{d,n}) /[n\log (n / {\lambda } )] \le C$ and $\mathrm{CT} ({\mathcal T}_{d,n})\le 3^{4\sqrt{ \log |{\cal V}_{d,n}| } }$.

Markov Chains and Mixing Times: Second Edition

Abstract: Unlike most books reviewed in the Intelligencer this is definitely a textbook. It assumes knowledge one might acquire in the first two years of an undergraduate mathematics program – basic mathematical probability, plus linear algebra, a little graph theory and the infamous concept of “mathematical maturity”. It has the theorem-proof style of pure mathematics, but with friendly explanations of intuition and motivation.

Uniform sampling through the Lovasz local lemma

Abstract: We propose a new algorithmic framework, called “partial rejection sampling”, to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds (perhaps surprising) new connections between the variable framework of the Lovasz Local Lemma and some clas- sical sampling algorithms such as the “cycle-popping” algorithm for rooted spanning trees by Wilson. Among other applications, we discover new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences.

Characterization of cutoff for reversible Markov chains

Abstract: A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least $\alpha$, for some $\alpha\in(0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some “worst” set of stationary measure at least $\alpha$ was not hit by time $t$. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to $\infty$.

A new lower bound for the critical probability of site percolation on the square lattice

Abstract: The critical probability for site percolation on the square lattice is not known exactly. Several authors have given rigorous upper and lower bounds. Some recent lower bounds are (each displayed here with the first three digits) 0.503 [Toth (1985)], 0.522 [Zuev (1988)] and, the best lower bound so far, 0.541 [Menshikov and Pelikh (1989)]. By a modification of the method of Menshikov and Pelikh we get a significant improvement, namely 0.556. Apart from a few classical results on percolation and coupling, which are explicitly stated in the Introduction, this paper is self-contained.