Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Free Random Variables

Nous etudions des facteurs M et N de type II 1 associes a de bonnes actions Bernoulli generalisees de groupes Γ et A ayant un sous-groupe infini presque-distingue avec la propriete (T) relative. Nous demontrons le resultat de rigidite suivant: chaque M-N-bimodule d'indice fini (en particulier, chaque isomorphisme entre M et N) peut etre decrit par une commensurabilite des groupes Γ, A et une commensurabilite de leurs actions. L'algebre de fusion des M-M-bimodules d'indice fini est identifiee avec une algebre de Hecke etendue, ce qui fournit les premiers calculs explicites de l'algebre de fusion d'un facteur de type II 1 . Nous obtenons en particulier des exemples explicites de facteurs II 1 dont l'algebre de fusion est triviale, ce qui veut dire que tous leurs sous-facteurs d'indice fini sont triviaux.

/pdf/explicit-computations-of-all-finite-index-bimodules-for-a-4n31onuhcl.pdf

Explicit computations of all finite index bimodules for a family of II$_1$ factors

We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely \remembers" the group G. More precisely, if LG is isomorphic to the von Neumann algebra L of an arbitrary countable group , then must be isomorphic to G. This represents the rst superrigidity result pertaining to group von Neumann algebras.

https://annals.math.princeton.edu/wp-content/uploads/annals-v178-n1-p04-s.pdf

A class of superrigid group von neumann algebras

We prove a “unique crossed product decomposition” result for group measure space II1 factors L ∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family $\mathcal{G}$
 , which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T
 n
 denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of $\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z})$
 is W∗-superrigid, i.e. any isomorphism between L ∞(X)⋊Γ and an arbitrary group measure space factor L ∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family $\mathcal{G}$
 , the Bernoulli actions of Γ are W∗-superrigid.

/pdf/group-measure-space-decomposition-of-ii1-factors-and-w-4n7sl6d0ft.pdf

Group measure space decomposition of II1 factors and W*-superrigidity

We obtain new Bass-Serre-type rigidity results for II1 equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any nonamenable factor arising as an amalgamated free product of von Neumann algebras M1*BM2 over an abelian von Neumann algebra B is prime, that is, cannot be written as a tensor product of diffuse factors. This gives, both in the type II1 and in the type III cases, new examples of prime factors.

Bass-Serre rigidity results in von Neumann algebras

https://www.imsc.res.in/~sunder/houdayer.pdf

Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors

Type III factors with unique Cartan decomposition

Unique prime factorization and bicentralizer problem for a class of type III factors

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A \rtimes \Gamma covering certain cases where \Gamma is an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if \Sigma < SL(3,\Z) denotes the subgroup of matrices g with g_31 = g_32 = 0, then any free ergodic probability measure preserving action of \Gamma = SL(3,\Z) *_\Sigma SL(3,\Z) is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.

https://www.ems-ph.org/journals/show_pdf.php?issn=1661-7207&vol=7&iss=3&rank=6

Cyril Houdayer

Papers

Bass-Serre rigidity results in von Neumann algebras

Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors

Type III factors with unique Cartan decomposition

Unique prime factorization and bicentralizer problem for a class of type III factors

A class of groups for which every action is W$^*$-superrigid