arXiv subjects
Publications and source records attributed to Antonio Auffinger.
We consider the fluctuations of the free energy of positive temperature directed polymers in thin rectangles (N,N^α), α< 3/14. For general weight distributions with finite fourth moment we prove that the distribution of these fluctuations converges as N goes to infinity to the GUE Tracy-Widom distribution.
We study first-passage percolation in two dimensions, using measures mu on passage times with b:=inf supp(mu) >0 and mu({b})=p \geq p_c, the threshold for oriented percolation. We first show that for each such mu, the boundary of the limit shape for mu is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if mu is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman-Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for mu. This result confirms a prediction of Newman-Piza and Zhang. Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents chi and xi.
We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power α, where α\in (0,2). After proper scaling of temperature β^{-1}, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (α, β)-indexed family of measures on Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal. In particular, this shows order n transversal fluctuations of the polymer. If, and only if, αis small enough, we find that there exists a random critical temperature below which, but not above, the effect of the environment is macroscopic. The results carry over to d+1 dimensions for d>1 with minor modifications.
We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.