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Itai Benjamini studies random geometric and combinatorial structures. His joint work with Russell Lyons, Yuval Peres and Oded Schramm investigates uniform spanning forests as limits of finite uniform spanning trees.^[ref]

On an infinite graph, a limiting forest need not be a single tree. Free and wired boundary conditions can lead to different laws. The paper relates this distinction and the number of connected components to random walks and harmonic functions, making infinite-volume geometry accessible through probabilistic tests.^[ref]

By David A. Levin and Yuval Peres, with contributions by Elizabeth L. Wilmer.

Author's book website: pages.uoregon.edu/dlevin/MARKOV/

Online text: pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf

Alternate author-hosted PDF: pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf

Publisher record: doi.org/10.1090/mbk/107

Cited passages: §4.1, Proposition 4.2 and Remark 4.3, printed p. 48: finite-state discrepancy formula; §4.2, Example 4.6, Proposition 4.7 and equation (4.13), printed pp. 49–52: coupling and common-mass construction. The author-hosted second-edition PDF text and title page were inspected; the numerical examples here are original.

Publication metadata: American Mathematical Society, 2017; hardcover ISBN 978-1-4704-2962-1, as listed in the publisher's ^[ref].

Russell Lyons studies probability theory on graphs. With Itai Benjamini, Yuval Peres and Oded Schramm, he developed the theory of uniform spanning forests on infinite graphs.^[ref]

Finite spanning-tree laws can approach different limits depending on whether the boundary is left free or wired together. Their work connects these limits with random walks, potential theory and harmonic functions. One result characterizes the number of wired-forest components through intersections of independent walks, showing that global connectivity is encoded in stochastic paths.^[ref]

Yuval Peres coauthored a treatment of Markov chains and mixing times. Coupling, spectral methods and probabilistic estimates quantify how quickly a chain approaches equilibrium, connecting a limiting distribution with the time needed to approximate it.^[ref]