Georgia Institute of Technology College of Sciences

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. We introduce a general framework of convex regularization for low rank tensor estimation.

In the 1970s, Girko made the striking observation that, after centering, traces of functions of large random matrices have approximately Gaussian distribution. This convergence is true without any further normalization provided f is smooth enough, even though the trace involves a number of terms equal to the dimension of the matrix. This is particularly interesting, because for some rougher, but still natural observables, like the number of eigenvalues in an interval, the fluctuations diverge. I will explain how such results can be obtained, focusing in particular on controlling the fluctuations when the function is not very regular.

In two-dimensional critical percolation, the work of Aizenman-Burchard implies that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+\epsilon, for some positive \epsilon. No more precise lower bound has been given so far. Conditioned on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Morrow and Zhang to have volume n^4/3 on the triangular lattice. In 1992, Kesten and Zhang asked how, given the existence of an open crossing, the length of the shortest open crossing compares to that of the lowest; in particular, whether the ratio of these lengths tends to zero in probability. We answer this question positively.