# On adding a list of numbers (and other one-dependent determinantal processes)

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Adding a column of numbers produces `carries' along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? (Many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae.) The examples give a gentle introduction to the emerging fields of one-dependent and determinantal point processes. This work is joint with Alexei Borodin and Persi Diaconis.

# Homogenization for non-coercive Hamilton-Jacobi equation.

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# Non-random perturbation of the Anderson Hamiltonian.

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# Intermittency for fully nonlinear equations

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# TBA

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# Large deviations from equilibrium measure for zeros of random holomorphic fields.

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An old result of Kac-Hammersley says that the complex zeros of a Gaussian

random polynomial \sum_{j = 0}^N a_j z^j with i.i.d. normal coefficients a_j, concentrate

on the unit circle. This seems counter intuitive at first, since the zeros could be anywhere.

We will explain this paradox and show that there is a very general result that empirical measures

of complex zeros tend to `equilibrium measures'. We then give a large deviations principle showing

that the probability of deviation from equilibrium measure is exponentially small.

# Perturbed simple random walk

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A random walk is called recurrent if it is sure to return to its starting point and transient otherwise. A famous result of Polya is that simple symmetric random walk on the integer lattice is recurrent in dimensions 1 and 2, and transient in higher dimensions. We study random walks that are small perturbations of simple random walk. Our main result is that if the dimension is high enough then these random walks are transient.

# "The Asymmetric Simple Exclusion Process: Integrable Structure and Limit Theorems"

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# Polymer Depinning Transitions with Loop Exponent One

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We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u+V_n when it visits a particular state 0 at time n, with V_n representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. Typically the probability of an excursion of length n for the underlying Markov chain is taken to decay as a power of n (called the loop exponent), perhaps with a slowly varying correction. A particular case not covered in a number of previous studies is that of loop exponent one, which includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. The work is joint with N. Zygouras.