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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 k th outlet and the volume of the k th pond. Next we prove bounds for all moments of(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)
J o u r n a l o f P r o b a b i l i t y Electron. Abstract We prove exponential concentration in i.i.d. first-passage percolation in Z d for all d ≥ 2 and general edge-weights (te). Precisely, under an exponential moment assumption (Ee αte < ∞ for some α > 0) on the edge-weight distribution, we prove the inequality P |T (0, x) − ET (0, x)| ≥ λ x1 log x1 ≤(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)
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 Z d for(More)
This two-day course will take place on Monday and Tues-day, January 2 and 3, before the Joint Meetings actually begin. It is co-organized by The objective of the course is to give an overview of recent exciting progress in the study of a class of stochastic growth models called first-and last-passage percolation (FPP and LPP). The issues involve limit(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)
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