arXiv subjects
Publications and source records attributed to Ariel Yadin.
We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z^d. The expected length of a random self avoiding path is considered. We also define a "critical exponent" $γ$ for sequences of graphs of size tending to infinity, and show that $γ= 1$ in the large girth case.
We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb{Z}^d$. We prove that the vector space of harmonic functions growing at most linearly is $(d+1)$-dimensional almost surely. Further, there are no nonconstant sublinear harmonic functions (thus implying the uniqueness of the corrector). A main ingredient of the proof is a quantitative, annealed version of the Avez entropy argument. This also provides bounds on the derivative of the heat kernel, simplifying and generalizing existing results. The argument applies to many different environments; even reversibility is not necessary.
We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.
In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.
We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting point is not more than a constant factor of the uniform measure on the set. For large sets there is a tight logarithmic correction factor. We also show that positive spectral gap does not allow for a fixed proportion of the harmonic measure of sets to be supported on small subsets, in contrast to the situation in Euclidean space. The results are quantitative as a function of the spectral gap, and apply also when the spectral gap decays to 0 as the size of the graph grows to infinity. As an application we consider a model of diffusion limited aggregation, or DLA, on finite graphs, obtaining upper bounds on the growth rate of the aggregate.
We study "tricolor percolation" on the regular tessellation of R^3 by truncated octahedra, which is the three-dimensional analog of the hexagonal tiling of the plane. We independently assign one of three colors to each cell according to a probability vector p = (p_1, p_2, p_3) and define a "tricolor edge" to be an edge incident to one cell of each color. The tricolor edges form disjoint loops and/or infinite paths. These loops and paths have been studied in the physics literature, but little has been proved mathematically. We show that each p belongs to either the compact phase (in which the length of the tricolor loop passing through a fixed edge is a.s. finite, with exponentially decaying law) or the extended phase (in which the probability that an n by n by n box intersects a tricolor path of diameter at least n exceeds a positive constant, independent of n). We show that both phases are non-empty and the extended phase is a closed subset of the probability simplex. We also survey the physics literature and discuss open questions, including the following: Does p=(1/3,1/3,1/3) belong to the extended phase? Is there a.s. an infinite tricolor path for this p? Are there infinitely many? Do they scale to Brownian motion? If p lies on the boundary of the extended phase, do the long paths have a scaling limit analogous to SLE_6 in two dimensions? What can be shown for the higher dimensional analogs of this problem?
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into $\mathbb{C}$ so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is $\mathrm{SLE}_2$. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of super-critical percolation on $\mathbb{Z}^2$. Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the loop-erased random walk on the super-critical percolation cluster is $\mathrm{SLE}_2$.
We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We show that the random trajectory becomes space-filling in the scaling limit when the parameter mu is supercritical.
Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2)).
Internal Diffusion Limited Aggregation (IDLA) is a model that describes the growth of a random aggregate of particles from the inside out. Shellef proved that IDLA processes on supercritical percolation clusters of integer-lattices fill Euclidean balls, with high probability. In this article, we complete the picture and prove a limit-shape theorem for IDLA on such percolation clusters, by providing the corresponding upper bound. The technique to prove upper bounds is new and robust: it only requires the existence of a "good" lower bound. Specifically, this way of proving upper bounds on IDLA clusters is more suitable for random environments than previous ways, since it does not harness harmonic measure estimates.
A k-wise independent distribution on n bits is a joint distribution of the bits such that each k of them are independent. In this paper we consider k-wise independent distributions with identical marginals, each bit has probability p to be 1. We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen as a relaxation of a major open problem in error-correcting codes theory, namely, how large can a linear error correcting code with given parameters be? The question is a type of discrete moment problem, and our approach is based on showing that bounds obtained from the theory of the classical moment problem provide good approximations for it. The main tool we use is a bound controlling the change in the expectation of a polynomial after small perturbation of its zeros.
We study the Laplacian-infinity path as an extreme case of the Laplacian-alpha random walk. Although, in the finite alpha case, there is reason to believe that the process converges to SLE, we show that this is not the case when alpha is infinite. In fact, the scaling limit depends heavily on the lattice structure, and is not conformal (or even rotational) invariant.
We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows that low-degree polynomial threshold functions cannot approximate any function with many influential variables. We provide a couple of examples where this technique yields tight approximation bounds. Our second result relates to constructing pseudorandom generators fooling low-degree polynomial threshold functions. This problem has received attention recently, where Diakonikolas et al proved that $k$-wise independence suffices to fool linear threshold functions. We prove that any low-degree polynomial threshold function, which can be represented as a function of a small number of linear threshold functions, can also be fooled by $k$-wise independence. We view this as an important step towards fooling general polynomial threshold functions, and we discuss a plausible approach achieving this goal based on our techniques. Our results combine tools from real approximation theory, hyper-contractive inequalities and probabilistic methods. In particular, we develop several new tools in approximation theory which may be of independent interest.
We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic factors).
We study the entropy of the set traced by an $n$-step random walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are essentially governed by the size of the boundary of the trace.
The mixer chain on a graph G is the following Markov chain. Place tiles on the vertices of G, each tile labeled by its corresponding vertex. A "mixer" moves randomly on the graph, at each step either moving to a randomly chosen neighbor, or swapping the tile at its current position with some randomly chosen adjacent tile. We study the mixer chain on Z, and show that at time t the expected distance to the origin is t^{3/4}, up to constants. This is a new example of a random walk on a group with rate of escape strictly between t^{1/2} and t.
In a function that takes its inputs from various players, the effect of a player measures the variation he can cause in the expectation of that function. In this paper we prove a tight upper bound on the number of players with large effect, a bound that holds even when the players' inputs are only known to be pairwise independent. We also study the effect of a set of players, and show that there always exists a "small" set that, when eliminated, leaves every set with little effect. Finally, we ask whether there always exists a player with positive effect. We answer this question differently in various scenarios, depending on the properties of the function and the distribution of players' inputs. More specifically, we show that if the function is non-monotone or the distribution is only known to be pairwise independent, then it is possible that all players have 0 effect. If the distribution is pairwise independent with minimal support, on the other hand, then there must exist a player with "large" effect.
Let A,B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group; i.e. (AB union BA) = G. The study of this event gives rise to a group invariant we call Θ(G). Θ(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes \sqrt{Θ(G)|G|\log|G|}; i.e. for any c>0, if the size of A and B is less than (1-c)\sqrt{Θ(G)|G|\log|G|}, then with high probability (AB union BA) does not equal G. If A and B are larger than (1+c)\sqrt{Θ(G)|G|\log|G|} then (AB union BA) equals G with high probability.