Abstract: We show that oriented percolation occurs whenever a condition is satisfied called" expo... more Abstract: We show that oriented percolation occurs whenever a condition is satisfied called" exponential intersection tails". This condition says that a measure on paths exists for which the probability of two independent paths intersecting in more than k sites is exponentially small in k.
Abstract How far can a stack of n identical blocks be made to hang over the edge of a table? The ... more Abstract How far can a stack of n identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order log n. However, at SODA'06, Paterson and Zwick constructed n-block stacks with overhangs of order n 1/3. Here we complete the solution to the overhang problem, and answer Paterson and Zwick's primary open question, by showing that order n 1/3 is best possible.
Abstract. We study the Glauber dynamics Markov chain fork-colourings of trees with maximum degree... more Abstract. We study the Glauber dynamics Markov chain fork-colourings of trees with maximum degree Δ. For k≥ 3, we show that the mixing time on every tree is at most nO (1+∆/(k log∆)). This bound is tight up to the constant factor in the exponent, as evidenced by the complete tree. Our proof uses a weighted canonical paths analysis and a variation of the block dynamics that exploits the differing relaxation times of blocks.
Abstract We consider a one-dimensional recurrent random walk in random environment (RWRE). We sho... more Abstract We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the–suitably centered–empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491–506.
Abstract: Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $ n $ in... more Abstract: Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $ n $ independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability $\ epsilon $, the probability $ p_\ epsilon $ that the weighted majority changes is at most $ C\ epsilon^{1/4} $. They asked what is the best possible exponent that could replace 1/4. We prove that the answer is 1/2.
Abstract We consider the fractal percolation process on the unit square with fixed decimation par... more Abstract We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {pk}; that is, for allk⩾ 1, at thek th stage every retained square of side lengthN 1− k is partitioned intoN 2 congruent subsquares, and each of these is retained with probabilityp k. independent of all others.
Abstract: We present two new methods for estimating the order (memory depth) of a finite alphabet... more Abstract: We present two new methods for estimating the order (memory depth) of a finite alphabet Markov chain from observation of a sample path. One method is based on entropy estimation via recurrence times of patterns, and the other relies on a comparison of empirical conditional probabilities. The key to both methods is a qualitative change that occurs when a parameter (a candidate for the order) passes the true order. We also present extensions to order estimation for Markov random fields.
Abstract We provide a complete description of the giant component of the Erdős-Rényi random graph... more Abstract We provide a complete description of the giant component of the Erdős-Rényi random graph equation image as soon as it emerges from the scaling window, ie, for p=(1+ ε)/n where ε 3 n→∞ and ε= o (1). Our description is particularly simple for ε= o (n-1/4), where the giant component equation image is contiguous with the following model (ie, every graph property that holds with high probability for this model also holds whp for equation image).
Abstract We study random walks on supercritical percolation clusters on wedges in Z3, and show th... more Abstract We study random walks on supercritical percolation clusters on wedges in Z3, and show that the infinite percolation cluster is (as) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on Z2 is equivalent to the (as) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman Key words: percolation, transience, wedges
Abstract Let S⊂(0, 1). Given a known function f: S→(0, 1), we consider the problem of using indep... more Abstract Let S⊂(0, 1). Given a known function f: S→(0, 1), we consider the problem of using independent tosses of a coin with probability of heads p (where p∈ S is unknown) to simulate a coin with probability of heads f (p). We prove that if S is a closed interval and f is real analytic on S, then f has a fast simulation on S (the number of p-coin tosses needed has exponential tails). Conversely, if a function f has a fast simulation on an open set, then it is real analytic on that set.
Abstract. We study models of spatial growth processes where initially there are sources of growth... more Abstract. We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no 'inter-green'competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as a red substance itself.
Abstract. The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a superc... more Abstract. The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an LlogL condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. Through a uni ed approach, we give conceptual proofs of these theorems which are free of manipulation of generating functions.
Abstract In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe... more Abstract In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β c, the inverse-gap is O (1) for β< β c, polynomial in the surface area for β= β c and exponential in it for β> β c. This has been proved for Z^ 2 except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph.
Abstract Let U⊆ C be a bounded domain with smooth boundary and let F be an instance of the contin... more Abstract Let U⊆ C be a bounded domain with smooth boundary and let F be an instance of the continuum Gaussian free field on U with respect to the Dirichlet inner product∫ U∇ f (x)⋅∇ g (x) dx. The set T (a; U) of a-thick points of F consists of those z∈ U such that the average of F on a disk of radius r centered at z has growth $\ sqrt {a/\ pi}\ log\ frac {1}{r} $ as r→ 0.
Every sufficiently large finite setX in [0, 1) has a dilationnX mod 1 with small maximal gap and ... more Every sufficiently large finite setX in [0, 1) has a dilationnX mod 1 with small maximal gap and even small discrepancy. We establish a sharp quantitative version of this principle, which puts into a broader perspective some classical results on the distribution of power residues. The proof is based on a second-moment argument which reduces the problem to an estimate on the number of edges in a certain graph.
Abstract We consider the Ising model on a general tree under various boundary conditions: all plu... more Abstract We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an L 3 capacity arises.
Abstract We show that if $ H $ is a group of polynomial growth whose growth rate is at least quad... more Abstract We show that if $ H $ is a group of polynomial growth whose growth rate is at least quadratic, then the $ L_p $ compression of the wreath product $\ mathbb {Z}\ bwr H $ equals\[{\ rm max}\ left\{\ frac {1}{p},\ frac {1}{2}\ right\}\].
Abstract We consider linear iterated function systems with a random multiplicative error on the r... more Abstract We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\ mapsto d_i+\ lambda_i Y x\} _ {i= 1}^ m $, where $ d_i\ in {\ mathbb R} $ and $\ lambda_i> 0$ are fixed and $ Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $ y_1, y_2,\ dotsc $, distributed as $ Y $, independent of everything else.
Abstract We prove that if U ⊂ R^ n is an open domain whose closure\ overline U is compact in the ... more Abstract We prove that if U ⊂ R^ n is an open domain whose closure\ overline U is compact in the path metric, and F is a Lipschitz function on∂ U, then for each β ∈ R there exists a unique viscosity solution to the β-biased infinity Laplacian equation β| ∇ u|+\ Delta_ ∞ u= 0 on U that extends F, where\ Delta_ ∞ u=| ∇ u|^-2 i, j u_ x_i u_ x_ix_j u_ x_j. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ϵ-game as follows. The starting position is x_0 ∈ U.
Abstract: We show that oriented percolation occurs whenever a condition is satisfied called" expo... more Abstract: We show that oriented percolation occurs whenever a condition is satisfied called" exponential intersection tails". This condition says that a measure on paths exists for which the probability of two independent paths intersecting in more than k sites is exponentially small in k.
Abstract How far can a stack of n identical blocks be made to hang over the edge of a table? The ... more Abstract How far can a stack of n identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order log n. However, at SODA'06, Paterson and Zwick constructed n-block stacks with overhangs of order n 1/3. Here we complete the solution to the overhang problem, and answer Paterson and Zwick's primary open question, by showing that order n 1/3 is best possible.
Abstract. We study the Glauber dynamics Markov chain fork-colourings of trees with maximum degree... more Abstract. We study the Glauber dynamics Markov chain fork-colourings of trees with maximum degree Δ. For k≥ 3, we show that the mixing time on every tree is at most nO (1+∆/(k log∆)). This bound is tight up to the constant factor in the exponent, as evidenced by the complete tree. Our proof uses a weighted canonical paths analysis and a variation of the block dynamics that exploits the differing relaxation times of blocks.
Abstract We consider a one-dimensional recurrent random walk in random environment (RWRE). We sho... more Abstract We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the–suitably centered–empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491–506.
Abstract: Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $ n $ in... more Abstract: Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $ n $ independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability $\ epsilon $, the probability $ p_\ epsilon $ that the weighted majority changes is at most $ C\ epsilon^{1/4} $. They asked what is the best possible exponent that could replace 1/4. We prove that the answer is 1/2.
Abstract We consider the fractal percolation process on the unit square with fixed decimation par... more Abstract We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {pk}; that is, for allk⩾ 1, at thek th stage every retained square of side lengthN 1− k is partitioned intoN 2 congruent subsquares, and each of these is retained with probabilityp k. independent of all others.
Abstract: We present two new methods for estimating the order (memory depth) of a finite alphabet... more Abstract: We present two new methods for estimating the order (memory depth) of a finite alphabet Markov chain from observation of a sample path. One method is based on entropy estimation via recurrence times of patterns, and the other relies on a comparison of empirical conditional probabilities. The key to both methods is a qualitative change that occurs when a parameter (a candidate for the order) passes the true order. We also present extensions to order estimation for Markov random fields.
Abstract We provide a complete description of the giant component of the Erdős-Rényi random graph... more Abstract We provide a complete description of the giant component of the Erdős-Rényi random graph equation image as soon as it emerges from the scaling window, ie, for p=(1+ ε)/n where ε 3 n→∞ and ε= o (1). Our description is particularly simple for ε= o (n-1/4), where the giant component equation image is contiguous with the following model (ie, every graph property that holds with high probability for this model also holds whp for equation image).
Abstract We study random walks on supercritical percolation clusters on wedges in Z3, and show th... more Abstract We study random walks on supercritical percolation clusters on wedges in Z3, and show that the infinite percolation cluster is (as) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on Z2 is equivalent to the (as) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman Key words: percolation, transience, wedges
Abstract Let S⊂(0, 1). Given a known function f: S→(0, 1), we consider the problem of using indep... more Abstract Let S⊂(0, 1). Given a known function f: S→(0, 1), we consider the problem of using independent tosses of a coin with probability of heads p (where p∈ S is unknown) to simulate a coin with probability of heads f (p). We prove that if S is a closed interval and f is real analytic on S, then f has a fast simulation on S (the number of p-coin tosses needed has exponential tails). Conversely, if a function f has a fast simulation on an open set, then it is real analytic on that set.
Abstract. We study models of spatial growth processes where initially there are sources of growth... more Abstract. We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no 'inter-green'competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as a red substance itself.
Abstract. The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a superc... more Abstract. The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an LlogL condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. Through a uni ed approach, we give conceptual proofs of these theorems which are free of manipulation of generating functions.
Abstract In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe... more Abstract In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β c, the inverse-gap is O (1) for β< β c, polynomial in the surface area for β= β c and exponential in it for β> β c. This has been proved for Z^ 2 except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph.
Abstract Let U⊆ C be a bounded domain with smooth boundary and let F be an instance of the contin... more Abstract Let U⊆ C be a bounded domain with smooth boundary and let F be an instance of the continuum Gaussian free field on U with respect to the Dirichlet inner product∫ U∇ f (x)⋅∇ g (x) dx. The set T (a; U) of a-thick points of F consists of those z∈ U such that the average of F on a disk of radius r centered at z has growth $\ sqrt {a/\ pi}\ log\ frac {1}{r} $ as r→ 0.
Every sufficiently large finite setX in [0, 1) has a dilationnX mod 1 with small maximal gap and ... more Every sufficiently large finite setX in [0, 1) has a dilationnX mod 1 with small maximal gap and even small discrepancy. We establish a sharp quantitative version of this principle, which puts into a broader perspective some classical results on the distribution of power residues. The proof is based on a second-moment argument which reduces the problem to an estimate on the number of edges in a certain graph.
Abstract We consider the Ising model on a general tree under various boundary conditions: all plu... more Abstract We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an L 3 capacity arises.
Abstract We show that if $ H $ is a group of polynomial growth whose growth rate is at least quad... more Abstract We show that if $ H $ is a group of polynomial growth whose growth rate is at least quadratic, then the $ L_p $ compression of the wreath product $\ mathbb {Z}\ bwr H $ equals\[{\ rm max}\ left\{\ frac {1}{p},\ frac {1}{2}\ right\}\].
Abstract We consider linear iterated function systems with a random multiplicative error on the r... more Abstract We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\ mapsto d_i+\ lambda_i Y x\} _ {i= 1}^ m $, where $ d_i\ in {\ mathbb R} $ and $\ lambda_i> 0$ are fixed and $ Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $ y_1, y_2,\ dotsc $, distributed as $ Y $, independent of everything else.
Abstract We prove that if U ⊂ R^ n is an open domain whose closure\ overline U is compact in the ... more Abstract We prove that if U ⊂ R^ n is an open domain whose closure\ overline U is compact in the path metric, and F is a Lipschitz function on∂ U, then for each β ∈ R there exists a unique viscosity solution to the β-biased infinity Laplacian equation β| ∇ u|+\ Delta_ ∞ u= 0 on U that extends F, where\ Delta_ ∞ u=| ∇ u|^-2 i, j u_ x_i u_ x_ix_j u_ x_j. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ϵ-game as follows. The starting position is x_0 ∈ U.
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