Michael Damron

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We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IICs). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the(More)
We study invasion percolation in two dimensions. We compare connectivity properties of the origin’s invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k ≥ 1 the k-point function of the first pond has the same asymptotic behaviour as the(More)
We prove exponential concentration in i.i.d. first-passage percolation inZ for all d ≥ 2 and general edge-weights (te). Precisely, under an exponential moment assumption (Eee <∞ for some α > 0) on the edge-weight distribution, we prove the inequality P ( |T (0, x)−ET (0, x)| ≥ λ √ ‖x‖1 log ‖x‖1 ) ≤ ce ′λ, ‖x‖1 > 1 for the point-to-point passage time T (0,(More)
We prove the scaling relation χ = 2ξ− 1 between the transversal exponent ξ and the fluctuation exponent χ for directed polymers in a random environment in d dimensions. The definition of these exponents is similar to that proposed in Chatterjee (2013) in first-passage percolation. The proof presented here also establishes the relation in the zero(More)
NOtices Of the AMs 1087 AMS Short Course on Random Growth Models This two-day course will take place on Monday and Tuesday, January 2 and 3, before the Joint Meetings actually begin. It is co-organized by Michael Damron, Georgia Institute of Technology, Firas Rassoul-Agha, University of Utah, and Timo Seppäläinen, University of Wisconsin–Madison. The(More)
Ground states of the Edwards-Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on Zd for(More)
Irregular and stochastic growth is all around us: tumors, bacterial colonies, infections, fluid spreading in a porous medium, propagating flame fronts. The study of simplified mathematical models of stochastic growth began in probability theory half a century ago. Quite serendipitously these models have turned out to be extremely hard to analyze. They have(More)
We introduce a new model for rill erosion. We start with a network similar to that in the Discrete Web** and instantiate a dynamics which makes the process highly non-Markovian. The behavior of nodes in the streams is similar to the behavior of Polya urns with time-dependent input. In this paper we use a combination of rigorous arguments and simulation(More)