Richard Durrett

Fixed points of the smoothing transformation

SummaryLet W1,..., WN be N nonnegative random variables and let

Intratumor Heterogeneity in Evolutionary Models of Tumor Progression

\mathfrak{M}

Evolution of resistance and progression to disease during clonal expansion of cancer

be the class of all probability measures on [0, ∞). Define a transformation T on

Graph fission in an evolving voter model

\mathfrak{M}

Spatial models for hybrid zones

by letting Tμ be the distribution of W1X1+ ... + WNXN, where the Xi are independent random variables with distribution μ, which are independent of W1,..., WN as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the Wi are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some γ>1, EWiγ<∞ for all i, we determine exactly when T has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, T always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett.

Random Walks, Brownian Motion, and Interacting Particle Systems

SummaryIn 1974, Mandelbrot introduced a process in [0, 1]2 which he called “canonical curdling” and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivity or “percolation” properties of these sets, proving all of the claims he made in Sect. 23 of the “Fractal Geometry of Nature” and a new one that he did not anticipate: There is a probability pc∈(0,1) so that if p<pc then the set is “duslike” i.e., the largest connected component is a point, whereas if p≧pc (notice the =) opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles. More succinctly put the system has a first order phase transition.