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We provide a general framework for computing mixing times of finite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. We introduce a new Markov chain on linear extensions of a poset with n vertices, which is a variant of the promotion Markov chain of Ayyer, Klee and the last author, and show that it has a mixing time O(n log n).
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Metastable Mixing of Markov Chains: Efficiently Sampling Low Temperature Exponential Random Graphs
In this paper we consider the problem of sampling from the low-temperature exponential random graph model (ERGM). The usual approach is via Markov chain Monte Carlo, but Bhamidi et al. showed that any local Markov chain suffers from an exponentially large mixing time due to metastable states. We instead consider metastable mixing, a notion of approximate mixing relative to the stationary distribution, for which it turns out to suffice to mix only within a collection of metastable states. We show that the Glauber dynamics for the ERGM at any temperature -- except at a lower-dimensional critical set of parameters -- when initialized at G(n,p) for the right choice of p has a metastable mixing time of O(n^2logn) to within total variation distance exp(−Ω(n)).
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- Award ID(s):
- 1940205
- PAR ID:
- 10479643
- Publisher / Repository:
- Institute of Mathematical Statistics
- Date Published:
- Journal Name:
- The Annals of applied probability
- ISSN:
- 1050-5164
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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