Abstract. We give an asymptotic evaluation of the complexity of spherical p-spin spinglass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our approach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of [BDG01].

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2 + log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension $N$, when $N$ is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.Comment: Published in at http://dx.doi.org/10.1214/13-AOP862 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Onétudie la loi des plus grandes valeurs propres de matrices aléatoires symétriques réelles et de covariance empirique quand les coefficients des matrices sontà queue lourde. Oń etend le résultat obtenu par A. Soshnikov dans [18] et on montre que le comportement asymptotique des plus grandes valeurs propres est déterminé par les plus grandes entrées de la matrice.Abstract. 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 A. Soshnikov in [18], we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

In 1979, G. Parisi [14] predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand [19] and later generalized to the mixed p-spin models by Panchenko [12]. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi