We prove upper and lower bounds on the free energy of the Sherrington-Kirkpatrick model with multidimensional spins in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional. We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle's probability cascades. For this purpose, an abstract quenched large deviations principle of the Gärtner-Ellis type is obtained. We derive Talagrand's representation of Guerra's remainder term for the Sherrington-Kirkpatrick model with multidimensional spins. The derivation is based on well-known properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which shed some light on the original convexity problem of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of a priori estimates.

Key words: random energy model, sums of random exponentials, zeros of random analytic functions, central limit theorem, extreme value theory, stable distributions, logarithmic potentials. AbstractThe partition function of the random energy model at inverse temperature β is a sum of random exponentials Z N (β) = N k=1 exp(β √ nX k ), where X 1 , X 2 , . . . are independent real standard normal random variables (= random energies), and n = log N . We study the large N limit of the partition function viewed as an analytic function of the complex variable β. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex β, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.

We study Derrida's generalized random energy model ͑GREM͒ in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by the strength of the external field and by parameters of the GREM.

We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and L 1 martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy. of Duisburg-Essen for hospitality. 1 = the same phase diagram as for the field of independent random energies. 2 = different phase diagram comparing to the REM one, due to the strictly larger leading order of the minimal energy than the one for the independent field of random energies. 1 arXiv:1704.05402v1 [math.PR]

We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complexvalued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase" -the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas [19], where the uncorrelated case was treated. In particular, our result covers the case of the real-valued BBM energy model at complex temperatures.The glassy phase of the complex temperature BBM Figure 1: Phase diagram of the REM (and conjecturally of the BBM energy model). The grey curves are the level lines of the limiting log-partition function, cf. (1.18). This paper mainly deals with phase B 2 .Koukiou [15]. The full phase diagram of this model at complex temperatures including the fluctuations and zeros of the partition function were identified by Kabluchko and one of us in [11]. In particular, the case of arbitrary correlations between the imaginary and real parts of the energies was considered in [11]. The same authors answered in [12] similar questions about the Generalized Random Energy model (GREM) -a model with hierarchical correlations -and obtained the full phase diagram. In the complex GREM, the phase diagram turned out to have a much richer structure than that of the complex REM. This sheds some light on the phase diagrams of the models beyond the complex REM universality class.(1.4) 1 This implies that p 0 = 0, so none of the particles ever dies. 2 The latter assumption is just a matter of normalization. Any expected number of children greater than 1 (= the supercritical regime) is allowed and the results of this paper remain valid with appropriate modifications of constants.3 Under the stated conditions, the convergence of the extremal process of BBM, on which we rely, is proven in [3]. For the case of branching random walk, using truncation techniques, Madaule [18] has shown the same under conditions that would in the Gaussian case imply finiteness of k p k k(ln k) 3 . This could probably be carried over to BBM. It is not clear whether the result holds under the Kesten-Stigum condition k p k k ln k < ∞. For a discussion on these issues, we refer to the lecture notes by Shi [23]. In the present paper, we are not concerned with improving the conditions on the offspring distribution.ECP 0 (2015), paper 0.