arXiv subjects

Robin Pemantle

Publications and source records attributed to Robin Pemantle.

At least 73 records · Page 4Linked to original sources

arXiv2004

Towards a theory of negative dependence

The FKG theorem says that the POSITIVE LATTICE CONDITION, an easily checkable hypothesis which holds for many natural families of events, implies POSITIVE ASSOCIATION, a very useful property. Thus there is a natural and useful theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few possible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.

math.PR↗

arXiv2004

Non-amenable products are not treeable

Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected components almost surely.

math.PR↗

arXiv2004

Where did the Brownian particle go?

Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W, started at the center of the sphere and run for unit time. Given the occupation measure mu of this projected path, what can be said about the terminal point W(1), or about the range of the original path? In any dimension, for each Borel set A subseteq S^{d-1}, the conditional probability that the projection of W(1) is in A given mu(A) is just mu(A). Nevertheless, in dimension d>=3, both the range and the terminal point of W can be recovered with probability 1 from mu. In particular, for d>=3 the conditional law of the projection of W(1) given mu is not mu. In dimension~2 we conjecture that the projection of W(1) cannot be recovered almost surely from mu, and show that the conditional law of the projection of W(1) given mu is not mu.

math.PR↗

arXiv2004

A phase transition in random coin tossing

Suppose that a coin with bias theta is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let mu_θbe the distribution of the observed sequence of coin tosses, and let u_n denote the chance of a renewal at time n. Harris and Keane showed that if sum_{n=1}^infty u_n^2=\infty, then mu_theta and μ_0 are singular, while if sum_{n=1}^{infty} u_n^2 theta_c, they are singular. We also prove that when u_n=O(n^{-1}), the measures mu_theta for theta in [-1,1] are all mutually absolutely continuous.

math.PR↗

arXiv2004

Uniform random spanning trees

There are several good reasons you might want to read about uniform spanning trees, one being that spanning trees are useful combinatorial objects. Not only are they fundamental in algebraic graph theory and combinatorial geometry, but they predate both of these subjects, having been used by Kirchoff in the study of resistor networks. This article addresses the question about spanning trees most natural to anyone in probability theory, namely what does a typical spanning tree look like?

math.PR↗

arXiv2004

Tree-indexed processes

This article examines a recent body of work on stochastic processes indexed by a tree. Emphasis is on the application of this new framework to existing probability models. Proofs are largely omitted, with references provided.

math.PR↗

arXiv2004

A Dynamic Model of Social Network Formation

We consider a dynamic social network model in which agents play repeated games in pairings determined by a stochastically evolving social network. Individual agents begin to interact at random, with the interactions modeled as games. The game payoffs determine which interactions are reinforced, and the network structure emerges as a consequence of the dynamics of the agents' learning behavior. We study this in a variety of game-theoretic conditions and show that the behavior is complex and sometimes dissimilar to behavior in the absence of structural dynamics. We argue that modeling network structure as dynamic increases realism without rendering the problem of analysis intractable.

math.PR↗

arXiv2004

Cycles in random k-ary maps and the poor performance of random random number generation

Knuth shows that iterations of a random function perform poorly on average as a random number generator. He proposes a generalization in which the next value depends on two or more previous values. This note demonstrates, via an analysis of the cycle length of a random k-ary map, the equally poor performance of a random instance in Knuth's more general model.

math.PR↗

arXiv2004

A probabilistic model for the degree of the cancellation polynomial in Gosper's Algorithm

Milenkovic and Compton in 2002 gave an analysis of the run time of Gosper's algorithm applied to a random input. The main part of this was an asymptotic analysis of the random degree of the cancellation polynomial c(k) under various stipulated laws for the input. Their methods use probabilistic transform techniques. Here, a more general classof input distributions is considered, and limit laws of the type proved by Milenkovic and Compton are shown to follow from a general functional central limit theorem. The methods herein are probabilistic and elementary and may be used to compute the means of the limiting distributions.

math.PR↗

arXiv2004

Network formation by reinforcement learning: the long and medium run

We investigate a simple stochastic model of social network formation by the process of reinforcement learning with discounting of the past. In the limit, for any value of the discounting parameter, small, stable cliques are formed. However, the time it takes to reach the limiting state in which cliques have formed is very sensitive to the discounting parameter. Depending on this value, the limiting result may or may not be a good predictor for realistic observation times.

math.PR↗

arXiv2004

Time to absorption in discounted reinforcement models

Reinforcement schemes are a class of non-Markovian stochastic processes. Their non-Markovian nature allows them to model some kind of memory of the past. One subclass of such models are those in which the past is exponentially discounted or forgotten. Often, models in this subclass have the property of becoming trapped with probability~1 in some degenerate state. While previous work has concentrated on such limit results, we concentrate here on a contrary effect, namely that the time to become trapped may increase exponentially in 1/x as the discount rate, 1-x, approaches~1. As a result, the time to become trapped may easily exceed the lifetime of the simulation or of the physical data being modeled. In such a case, the quasi-stationary behavior is more germane. We apply our results to a model of social network formation based on ternary (three-person) interactions with uniform positive reinforcement.

math.PR↗

arXiv2004

Vertex-Reinforced Random Walk

This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix, R, which is real, symmetric and nonnegative. Let S_i(n) keep track of the number of visits to state i up to time n, and form the fractional occupation vector, V(n), where v_i(n)=S_i(n)/(sum_{j=1}^d S_j(n)). It is shown that V(n) converges to a set of critical points for the quadratic form H with matrix R, and that under nondegeneracy conditions on R, there is a finite set of points such that with probability one, V(n)->p for some p in the set. There may be more than one p in this set for which P(V(n)->p)>0. On the other hand P(V(n)->p)=0 whenever p fails in a strong enough sense to be maximum for H.

math.PR↗

arXiv2004

Critical RWRE on trees and tree-indexed random walks

We study the behavior of Random Walk in Random Environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree Gamma are i.i.d.random variables whose logarithms have mean zero and finite variance. Then the resulting RWRE is transient if simple random walk on Gamma is transient, but not vice versa. We obtain general transience criteria for such walks, which are sharp for symmetric trees of polynomial growth. In order to prove these criteria, we establish results on boundary crossing by tree-indexed random walks. These results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean-zero finite variance random walks in one dimension, which are of independent interest.

math.PR↗

arXiv2004

Choosing a Spanning Tree for the Integer Lattice Uniformly

Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for Z^d. This is shown to be a tree if and only if d=<4. In this case, the tree has only one topological end, i.e. there are no doubly infinite paths. When d>=5 the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

math.PR↗

arXiv2004

Domination Between Trees and Application to an Explosion Problem

We define a notion of stochastic domination between trees, where one tree dominates another if when the vertices of each are labeled with independent, identically distributed random variables, one tree is always more likely to contain a path with a specified property. Sufficient conditions for this kind of domination are (1) more symmetry and (2) earlier branching. We apply these conditions to the problem of determining how fast a tree must grow before first-passage percolation on the tree exhibits an explosion, that is to say, infinitely many vertices are reached in finite time. For a tree in which each vertex at distance n-1 from the root has f(n) offspring, f nondecreasing, an explosion occurs with exponentially distributed passage times if and only if sum f(n)^{-1}<infty.

math.PR↗

arXiv2004

Random Walk in a Random Environment and First-Passage Percolation on Trees

We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is likewise determined by the branching number of the tree. Finally, we show that the branching number determines the rate of first-passage percolation on trees, also known as the first-birth problem. Our techniques depend on quasi-Bernoulli percolation and large deviation results.

math.PR↗

arXiv2004

The Contact Process on Trees

The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter lambda is varied. For small values of lambda a single infection eventually dies out. For larger lambda the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of lambda, and the proof of this is much easier than it is for the contact process on d-dimensional integer lattices.) For still larger lambda the infection converges in distribution to a nontrivial invariant measure. For an n-ary tree, with n large, the first of these transitions occurs when lambda~1/n and the second occurs when 1/2 sqrt{n}<lambda<e/sqrt{n}. Nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n-ary trees, provided that vertices of degree n are not too rare. In particular, letting n go to infty, Galton-Watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at lambda=0. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.

math.PR↗

arXiv2004

On Path Integrals for the High-Dimensional Brownian Bridge

Let v be a bounded function with bounded support in R^d, d>=3. Let x,y in R^d. Let Z(t) denote the path integral of v along the path of a Brownian bridge in R^d which runs for time t, starting at x and ending at y. As t->infty, it is perhaps evident that the distribution of Z(t) converges weakly to that of the sum of the integrals of v along the paths of two independent Brownian motions, starting at x and y and running forever.Here we prove a stronger result, namely convergence of the corresponding moment generating functions and of moments. This result is needed for applications in physics.

math.PR↗