The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are expressed through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisi's ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue.PACS numbers: 75.50.Lk IntroductionThe statistical mechanics of spin-glass models is characterized by the existence of a diverse collection of competing states, very slow relaxation of the quenched dynamics, and a rather involved picture of the equilibrium state.A great deal of insight on the subject has been produced through the study of the Sherrington Kirkpatrick (SK) model [1]. After some initial attempts, a solution was proposed by G. Parisi which has the requisite stability and many other attractive features [2]. Its development has yielded a plethora of applications of the method, in which a key structural assumption is a particular form of the replica symmetry-breaking (i.e., the assumption of "ultrametricity", or the hierarchal structure, of the overlaps among the observed spin configurations) [3].Yet to this day it was not established that this very appealing proposal does indeed provide the equilibrium structure of the SK model. A recent breakthrough is the proof by F. Guerra [4] that the free energy provided by Parisi's purported solution is a rigorous lower bound for the SK free energy.More completely, the result of Guerra is that for any value of the order parameter, which within the assumed ansatz is a function, the Parisi functional provides a rigorous lower bound. Thus, this relation is also valid for the maximizer which yields the Parisi solution.In this work we present a variational principle for the free energy of the SK model which makes no use of a Parisi-type order parameter, and which yields the result of Guerra as a particular implication. More explicitly, the new principle allows more varied bounds on the free energy, for which there is no need to assume a hierarchal organization of the Gibbs state (e.g., as expressed in the assumed ultrametricity of the overlaps [3]). Guerra's results follow when the variational principle is tested against the Derrida-Ruelle hierarchal probability cascade models (GREM) [5].This leads us to a question which is not new: is the ultrametricity an inherent structue of the SK mean-field model, or is it only a simplifiying assumption. The new

ABSTRACT. The Sherrington-Kirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly diversified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical study of its free energy. In particular, we highlight the perspective of the cavity dynamics, and the related variational principle. These are expressed in terms of Random Overlap Structures (ROSt), which are used to describe the possible states of the reservoir in the cavity step. The Parisi solution is presented as reflecting the ansatz that it suffices to restrict the variation to hierarchal structures which are discussed here in some detail. While the Parisi solution was proven to be correct, through recent works of F. Guerra and M. Talagrand, the reasons for the effectiveness of the Parisi ansatz still remain to be elucidated. We question whether this could be related to the quasi-stationarity of the special subclass of ROSts given by Ruelle's hierarchal 'random probability cascades' (also known as GREM). CONTENTS

We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy A. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large A, about its overall behavior as a function of A and its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum.e-print archive: http://xxx.lanLgov/math-ph/0110017 0 Copyright © 2001 by the authors. Reproduction of this article in its entirety, by any means, is permitted for non-commercial purposes. 1048Koma, Nachtergaele, and Starr By combining information obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a Solid-on-Solid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.

Abstract. The Mallows model on Sn is a probability distribution on permutations, q d(π,e) /Pn(q), where d(π, e) is the distance between π and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs (i, j) where 1 ≤ i < j ≤ n, but π i > π j . Analyzing the normalization Pn(q), Diaconis and Ram calculated the mean and variance of d(π, e) in the Mallows model, which suggests the appropriate n → ∞ limit has qn scaling as 1 − β/n. We calculate the distribution of the empirical measure in this limit,Treating it as a mean-field problem, analogous to the Curie-Weiss model, the self-consistent mean-field equations are

The Mallows measure on the symmetric group Sn is the probability measure such that each permutation has probability proportional to q raised to the power of the number of inversions, where q is a positive parameter and the number of inversions of π is equal to the number of pairs i < j such that πi > πj. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that n → ∞ and q → 1 in such a way that n(1 − q) has a limit in R.