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

We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism (under group action) (SI) and Coset Intersection (CI) can be solved in quasipolynomial (exp (logn) O(1) � ) time. The best previous bound for GI was exp(O( √ nlogn)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp( e O( √ n)), where n is the size of the permutation domain (Babai, 1983). The algorithm builds on Luks’s SI framework and attacks the barrier configurations for Luks’s algorithm by group theoretic “local certificates” and combinatorial canonical partitioning techniques. We show that in a well-defined sense, Johnson graphs are the only obstructions to effective canonical partitioning.

Graph Isomorphism in Quasipolynomial Time

We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) andCoset Intersection (CI) can be solved in quasipolynomial(exp((logn)O(1))) time. The best previous bound for GI was exp(O( √n log n)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(O~(√ n)), where n is the size of the permutation domain (Babai, 1983). Following the approach of Luks’s seminal 1980/82 paper, the problem we actually address is SI. This problem takes two strings of length n and a permutation group G of degree n (the “ambient group”) as input (G is given by a list of generators) and asks whether or not one of the strings can be transformed into the other by some element of G. Luks’s divide-and-conquer algorithm for SI proceeds by recursion on the ambient group. We build on Luks’s framework and attack the obstructions to efficient Luks recurrence via an interplay between local and global symmetry. We construct group theoretic “local certificates” to certify the presence or absence of local symmetry, aggregate the negative certificates to canonical k-ary relations where k = O(log n), and employ combinatorial canonical partitioning techniques to split the k-ary relational structure for efficient divide-and- conquer. We show that in a well–defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. The central element of the algorithm is the “local certificates” routine which is based on a new group theoretic result, the “Unaffected stabilizers lemma,” that allows us to construct global automorphisms out of local information.

Graph isomorphism in quasipolynomial time [extended abstract]

Let G be a locally compact (second countable) group and π:G → U(ℋ) a unitary representation of G on the (separable) Hilbert space ℋ.Of course,ℋ may or may not contain non-trivial vectors invariant under π(G). We wish to define a notion of π almost containing invariant vectors.

Kazhdan’s Property (T)

Leçons sur l'intégration et la recherche des fonctions primitives professées au Collège de France: TABLE DES MATIÈRES

Main theorems of the article concern the problem of Atiyah on possible values of $$l^2$$
 -Betti numbers. It is shown that all non-negative real numbers are $$l^2$$
 -Betti numbers, and that “many” (for example all non-negative algebraic) real numbers are $$l^2$$
 -Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental $$l^2$$
 -Betti number with respect to an action of the threefold direct product of the lamplighter group $$\,\mathbb {Z}{/2\mathbb {Z}}\,\wr \mathbb {Z}$$
 . The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.

On Turing dynamical systems and the Atiyah problem

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.

Given an action of a group $\Gamma$ on a measure space $\Omega$, we provide a sufficient criterion under which two sets $A, B\subseteq \Omega$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable pieces which can be rearranged using the elements of $\Gamma$ to form a partition of $B$. In particular, we prove that every bounded measurable subset of $R^n$, $n\ge 3$, with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension $n\ge 2$.

/pdf/measurable-equidecompositions-for-group-actions-with-an-3yrbmlk2db.pdf

Measurable equidecompositions for group actions with an expansion property

We prove a Borel version of the local lemma, ie we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying assignment which is a Borel function The main tool which we develop for the proof, which is of independent interest, is a parallel version of the Moser-Tardos algorithm which uses the same random bits to resample clauses that are far enough in the dependency graph

Borel version of the Local Lemma

Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third problem.

/pdf/measurable-circle-squaring-drf231jzti.pdf

Łukasz Grabowski

Papers

On Turing dynamical systems and the Atiyah problem

On Turing dynamical systems and the Atiyah problem

Measurable equidecompositions for group actions with an expansion property

Borel version of the Local Lemma

Measurable circle squaring