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

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... ariable. 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. the process gets more and more de-correlated, and in the limit p " +1 we a r r i v e at the case where X are i.i.d. normal random variables. 4 The overlap is related to the Hamming distance d Ha m by d Ha m ( 0 ) = N(1 ; R N ( 0 ))=2. 5 The case p odd can also be treated, but presents considerable additional computational problems.