The method of exchangeable pairs can also be used to analyze nonlinear matrix-valued functions of random variables. In this section, we explain how to analyze matrix functions that satisfy a self-reproducing property.

Limiting shape. In the two papers [13, 14] the authors studied the limiting shape of the low temperature 2d Ising with minus boundary under a prescribed small positive external field, proportional to the inverse of the side-length L. The behavior of…

We thank anonymous referees for helpful comments.

Implicit in Theorem 1.2 is the fact that for any distribution of H on (0,1) with μlog:=E−log(1−H)<∞ and −log(1−H) non-lattice, that for the Markov chain Q∙ with mixed negative binomial transition matrix (1.17) derived from moments of H, with initial…

In this section we again consider the case that q=2 and majority dynamics (i.e., each voter adopts the majority opinion of its neighbors), with a population wide majority vote at time T. To avoid the issue of ties, we assume that |Nv| is odd for all…

The authors would like to thank Jeffrey Steif for helpful comments on an earlier draft.

We are grateful to David Aldous, Riddhipratim Basu and Allan Sly for many useful discussions. In addition we want to thank Riddhipratim Basu and Emma Cohen for reading previous drafts of this work and suggesting many improvements to the presentation.

The basic observation underlying our approach is that distances between populations may be estimated correctly using the minimum divergence time among all individuals and all genes.