Yuval Peres

Probability on Trees and Networks

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.

Determinantal Processes and Independence

We give a probabilistic introduction to determinantal and per- manental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L 2 (D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental pro- cesses, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

Sixty Years of Bernoulli Convolutions

The distribution νλ of the random series random series Σ±λn is the infinite convolution product of These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which λ∈ is νλ, absolutely continuous?

Rigorous location of phase transitions in hard optimization problems

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme ‘hardness’, computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

The threshold for random k-SAT is 2 k (ln 2 - O(k))

Let <i>F_k(n,m)</i> be a random <i>k</i>-SAT formula on <i>n</i> variables formed by selecting uniformly and independently <i>m</i> out of all possible <i>k</i>-clauses. It is well-known that for <i>r ≥ 2^k ln 2</i>, <i>F_k(n,rn)</i> is unsatisfiable with probability <i>1-o(1)</i>. We prove that there exists a sequence <i>t_k = O(k)</i> such that for <i>r ≥ 2^k ln 2 - t_k</i>, <i>F_k(n,rn)</i> is satisfiable with probability <i>1-o(1)</i>.Our technique yields an explicit lower bound for every <i>k</i> which for <i>k > 3</i> improves upon all previously known bounds. For example, when <i>k=10</i> our lower bound is 704.94 while the upper bound is 708.94.

Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions

Erdős (1939, 1940) studied the distribution νλ of the random series P∞ 0 ±λn, and showed that νλ is singular for infinitely many λ ∈ (1/2, 1), and absolutely continuous for a.e. λ in a small interval (1 − δ, 1). Solomyak (1995) proved a conjecture made by Garsia (1962) that νλ is absolutely continuous for a.e. λ ∈ (1/2, 1). In order to sharpen this result, we have developed a general method that can be used to estimate the Hausdorff dimension of exceptional parameters in several contexts. In particular, we prove: • For any λ0 > 1/2, the set of λ ∈ [λ0, 1) such that νλ is singular has Hausdorff dimension strictly less than 1. • For any Borel set A ⊂ Rd with Hausdorff dimension dim A > (d + 1)/2, there are points x ∈ A such that the pinned distance set {|x− y| : y ∈ A} has positive Lebesgue measure. Moreover, the set of x where this fails has Hausdorff dimension at most d + 1− dim A. • Let Kλ denote the middle-α Cantor set for α = 1 − 2λ and let K ⊂ R be any compact set. Peres and Solomyak (1998) showed that for a.e. λ ∈ (λ0, 1/2) such that dim K + dimKλ > 1, the sum K + Kλ has positive length; we show that the set of exceptional λ in this statement has Hausdorff dimension at most 2− dim K − dimKλ0 . • For any Borel set E ⊂ Rd with dim E > 2, almost all orthogonal projections of E onto lines through the origin have nonempty interior, and the exceptional set of lines where this fails has dimension at most d + 1− dim E. • If μ is a Borel probability measure on Rd with correlation dimension greater than m + 2γ, then for a “prevalent” set of C1 maps f : Rd → Rm (in the sense described by Hunt, Sauer and Yorke (1992)), the image of μ under f has a density with at least γ fractional derivatives in L2(Rm).

Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure

We consider simple random walk on the family tree T of a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary of T. Concretely, this implies that an exponentially small fraction of the nth level of T carries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.

Tug-of-war with noise: A game-theoretic view of the

Fix a bounded domain Ω ⊂ Rd, a continuous function F : ∂Ω → R, and constants ǫ > 0 and 1 < p, q < ∞ with p−1 + q−1 = 1. For each x ∈ Ω, let uǫ(x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v ∈ B(0, ǫ) to add to the game position, after which a random “noise vector” with mean zero and variance q p |v| 2 in each orthogonal direction is also added. The game ends when the game position reaches some y ∈ ∂Ω, and player I’s payoff is F (y). We show that (for sufficiently regular Ω) as ǫ tends to zero the functions uǫ converge uniformly to the unique p-harmonic extension of F . Using a modified game (in which ǫ gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F . These games and their variants interpolate between the tug of war games studied by Peres, Schramm, Sheffield, and Wilson (p = ∞) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p = 1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure.

p

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

-Laplacian

1. Preface 2. Basic Definitions and a Few Highlights 3. Galton-Watson Trees 4. General percolation on a connected graph 5. The First-Moment Method 6. Quasi-independent Percolation 7. The Second Moment Method 8. Electrical Networks 9. Infinite Networks 10. The Method of Random Paths 11. Transience of Percolation Clusters 12. Subperiodic Trees 13. The Random Walks \({\rm RW}_\lambda\) 14. Capacity 15. Intersection-Equivalence 16. Reconstruction for the Ising Model on a Tree 17. Unpredictable Paths in Z and EIT inZ2 18. Tree-Indexed Processes 19. Recurrence for Tree-Indexed Markov Chains 20. Dynamical Percolation 21. Stochastic Domination Between Trees