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Robin Pemantle

Publications and source records attributed to Robin Pemantle.

88 records · Page 5Linked to original sources
arXiv2004

Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings via Transfer-Impedances

Let G be a finite graph or an infinite graph on which Z^d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, known methods show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finite-dimensional marginals) of T from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix (n-dimensional minors) are needed to compute small (n-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for G, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when G is an infinite graph admitting a Z^d-action with finite quotient. The same computation also gives the entropy of the law of T. These results are applied to the problem of tiling certain lattices by dominos - the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to tree-valued random variables and criteria for these generalized moments to determine a distribution.

math.PR↗
arXiv2004

Critical Random Walk in Random Environment on Trees of Exponential Growth

This paper studies the behavior of RWRE on trees in the critical case left open in previous work. For trees of exponential growth, a random perturbation of the transition probabilities can change a transient random walk into a recurrent one. This is the opposite of what occurs on trees of sub-exponential growth.

math.PR↗
arXiv2004

Planar first-passage percolation times are not tight

We consider first-passage percolation on the two-dimensional integer lattice Z^2 with passage times that are IID exponentials of mean one. It has been conjectured, based on numerical evidence, that the variance of the time T(0,n) to reach the vertex (0,n) is of order n^{2/3}. Kesten showed that the variance of T(0,n) is at O(n). He also noted that the variance is bounded away from zero. This note improves the lower bound on the variance of T(0,n) to C log n. Simultaneously and independently, Newman and Piza have achieved the same result for {0,1}-valued passage times. Their methods extend to more general passage times, while ours work only for exponential times. On the other hand, our theorem shows that the variance comes from fluctuations of nonvanishing probability in the sense that, as n->infty, the law of T(0,n) is not tight about its median. Very recently, Newman and Piza showed that the log n may be improved to a power of n for directions in which the shape is not flat (it is not known whether the shape can be flat in any direction). As pointed out to us by Harry Kesten, in the exponential case this may also be obtained via the method given here.

math.PR↗
arXiv2004

A Shuffle that Mixes Sets of any Fixed Size much Faster than it Mixes the Whole Deck

Consider an n by n array of cards shuffled in the following manner. An element x of the array is chosen uniformly at random; Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180 degrees, and with probability 1/2 the rectangle of cards below and to the right of x is rotated 180 degrees. It is shown by an eigenvalue method that the time required to approach the uniform distribution is between n^2/2 and cn^2 ln n for some constant c. On the other hand, for any k it is shown that the time needed to uniformly distribute a set of cards of size k is at most c(k)n, where c(k) is a constant times k^3 ln(k)^2. This is established via coupling; no attempt is made to get a good constant.

math.PR↗
arXiv2004

Galton-Watson Trees with the Same Mean Have the Same Polar Sets

Evans defines a notion of what it means for a set B to be polar for a process indexed by a tree. The main result herein is that a tree picked from a Galton-Watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it follows that certain random Cantor sets are capacity-equivalent to each other and to deterministic Cantor sets. An extension to branching processes in varying environment is also obtained.

math.PR↗
arXiv2004

Martin Capacity for Markov Chains

The probability that a transient Markov chain, or a Brownian path, will ever visit a given set Lambda, is classically estimated using the capacity of Lambda with respect to the Green kernel G(x,y). We show that replacing the Green kernel by the Martin kernel G(x,y)/G(0,y) yields improved estimates, which are exact up to a factor of 2. These estimates are applied to random walks on lattices, and also to explain a connection found by R. Lyons between capacity and percolation on trees.

math.PR↗
arXiv2004

Percolation, first-passage percolation, and covering times for Richardson's model on the n-cube

Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube B_n ={0,1}^n with nearest neighbor edges. For oriented and unoriented percolation, p=e/n and p=1/n are the respective critical probabilities. For oriented first-passage percolation with i.i.d. edge-passage times having a density of 1 near the origin, the percolation time (time to reach the opposite corner of the cube) converges in probability to 1 as n->infty. This resolves a conjecture of David Aldous. When the edge-passage distribution is standard exponential, the (smaller) percolation time for unoriented edges is at least 0.88. These results are applied to Richardson's model on the (unoriented) n-cube. Richardson's model, otherwise known as the contact process with no recoveries, models the spread of infection as a Poisson process on each edge connecting an infected node to an uninfected one. It is shown that the time to cover the entire n-cube is bounded between 1.41 and 14.05 in probability as n->infty.

math.PR↗
arXiv2002

Generating a random sink-free orientation in quadratic time

A sink-free orientation of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately from the uniform distribution on sink-free orientations in time O(m^3 log (1/epsilon)), where m is the number of edges and epsilon the degree of approximation. Huber (1998) uses coupling from the past to obtain an exact sample in time O(m^4). We present a simple randomized algorithm inspired by Wilson's cycle popping method which obtains an exact sample in mean time at most O(nm), where n is the number of vertices.

math.PR↗
arXiv1997

Markov chains in a field of traps

A general criterion is given for when a Markov chain trapped with probability p(x) in state x will be almost surely trapped. The quenched (state x is a trap forever with probability p(x)) and annealed (state x traps with probability p(x) on each visit) problems are shown to be equivalent.

math.PR↗
arXiv1997

Vertex-reinfoced random walk on Z visits finitely many states

Vertex-reinforced random walk is defined in Pemantle's (1988) thesis; it is a random walk that is biased to visit sites it has already visited a lot. We show that this reinforcement scheme, in contrast to the scheme of edge-reinforcement, causes random walk on a line to get trapped in a finite set.

math.PR↗
arXiv1997

Sets avoided by Brownian motion

Any fixed cylinder is hit almost surely by a 3-dimensional Brownian motion, but is there a random cylinder that is in the complement? We answer this for cylinders, and then replacing a cylinder with a more general set.

math.PR↗
arXiv1997

Paths with exponential intersection tails and oriented percolation

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.

math.PR↗
arXiv1997

No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1.

math.PR↗
arXiv1997

The probability that Brownian motion almost covers a line

Lower and upper estimates are given for the probability that the epsilon-enlargement of planar Brownian motion to time 1 (the epsilon sausage) contains a unit line segment. The estimates imply that Brownian motion to time 1 itself contains no line segment.

math.PR↗
arXiv1995

The dimension of the Brownian frontier is greater than 1

Consider a planar Brownian motion run for finite time. The frontier or ``outer boundary'' of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension $4/3$, but this is still open.) The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands.

math.PR↗