We consider the random uctuations of the free energy in the p-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a r e c e n t paper by T alagrand on the p-spin version, we prove that (for p even) the random corrections to the free energy are on a scale N ;(p;2)=4 only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, , smaller than a critical p . We also show that p ! p 2 l n 2 a s p " +1. Additionally we study the formal p " +1 limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at all temperatures. For < p 2 l n 2 , uctuations are found at an exponentially small scale, with two distinct limit laws above and below a second critical value p ln 2=2: For up to that value the rescaled uctuations are Gaussian, while below that there are non-Gaussian uctuations driven by t h e P oisson process of the extreme values of the random energies. For larger than the critical p 2 l n 2 , the uctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1=2.

Subject Classification: 62E20, 62H20, 62P05

Abstract. In [7] Krotov and Hopfield suggest a generalized version of the wellknown Hopfield model of associative memory. In their version they consider a polynomial interaction function and claim that this increases the storage capacity of the model. We prove this claim and take the "limit" as the degree of the polynomial becomes infinite, i.e. an exponential interaction function. With this interaction we prove that model has an exponential storage capacity in the number of neurons, yet the basins of attraction are almost as large as in the standard Hopfield model.

We derive the upper‐tail moderate deviations for the length of a longest increasing subsequence in a random permutation. This concerns the regime between the upper‐tail large‐deviation regime and the central limit regime. Our proof uses a formula to describe the relevant probabilities in terms of the solution of the rank 2 Riemann‐Hilbert problem (RHP); this formula was invented by Baik, Deift, and Johansson [3] to find the central limit asymptotics of the same quantities. In contrast to the work of these authors, who apply a third‐order (nonstandard) steepest‐descent approximation at an inflection point of the transition matrix elements of the RHP, our approach is based on a (more classical) second‐order (Gaussian) saddle point approximation at the stationary points of the transition function matrix elements. © 2001 John Wiley & Sons, Inc.

We analyze the spectral distribution of symmetric random matrices with correlated entries. While we assume that the diagonals of these random matrices are stochastically independent, the elements of the diagonals are taken to be correlated. Depending on the strength of correlation the limiting spectral distribution is either the famous semicircle law or some other law, related to that derived for Toeplitz matrices by Bryc, Dembo and Jiang (2006).