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We review recent theoretical progress on glassy dynamics, with special emphasis on the importance and universality of the ``aging regime'', which is relevant to many experimental situations. The three main subjects which we address are: (i) Phenomenological models of aging (coarsening, trap models), (ii) Analytical results for the low-temperature dynamics of mean-field models (corresponding to the mode-coupling equations); and (iii) Simple non-disordered models with glassy dynamics. We discuss the interrelation between these approaches, and also with previous work in the field. Several open problems are underlined -- in particular the precise relation between mean-field like (or mode-coupling) descriptions and finite dimensional problems.

We study a system of Ismg spins with quenched random mfimte ranged p-spin interactions For p -* co, we can solve thts model exactly either by a direct mlcrocanomcal argument, or through the introduction of rephcas and Pansl's ultrametnc ansatz for rephca symmetry breaking, or by means of TAP mean field equat,ons Although the model is extremely simple it retains the charactertstlc features of a spin glass We use ~t to confirm the methods that have been apphed in more comphcated sxtuatlons and to exphcltly exhibit the structure of the spin glass phase

We use the replica method to study the (bipartite) weighted matching problem with independent random distances between the points. We propose a replica symmetric solution which fits the numerical values of the minimal length and the distribution of lengths of the occupied links in the optimal configuration Nous utilisons la methode des repliques pour etudier le probleme du matching (bipartite) avec des distances aleatoires independantes entre les points. Nous proposons une solution, symetrique dans les repliques, qui est en accord avec les valeurs numeriques de la longueur minimale et la distribution des longueurs des liens occupes dans la configuration optimale.

The hierarchical organization of the pure states of a S.K. spin glass (ultrametricity) is analysed in terms of self-averaging distributions of local magnetizations. We show that every pure state α defines an ultrametric distance Dα(i,j) among the N sites. Given two states α, β with overlap q there is a minimum distande d m such that for two sites i, j with Dα(i,j)≥d m the two distances Dα and Dβ coincide. It follows that the sites can be partitioned in disjoint cells inside which the total magnetization is the same for all the states with mutual overlap q. For this same family of states we then define an «ancestor» that has, inside each cell, constant local magnetization equal to the average magnetization of the descendants. The ancestors satisfy mean field like equations. The functional dependence of the local magnetization in terms of the local field is given by the solution of the diffusion equation in x space which is given a purely static interpretation Analyse de l'organisation hierarchique des etats purs d'un verre de spin de Sherrington-Kirkpatrick en termes de la distribution des aimantations locales: chaque etat pur α definit une distance ultrametrique Dα(i,j) entre les N sites et, etant donnes deux etats α, β dont le recouvrement est q, il y a une distance minimale d m telle que, pour chaque paire de site i, j verifiant Dα(i,j)≥d m , les deux distances Dα et Dβ coincident

A probability distribution has been proposed recently by one of us as an order parameter for spin glasses. We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution. We also show that the space of states has an ultrametric topology Etude d'une distribution de probabilite, proposee recemment, comme parametre d'onde pour les verres de spin: la probabilite depend de la realisation particuliere des couplages, meme a la limite thermodynamique. Etude de la distribution de cette probabilite. Mise en evidence de la topologie ultrametrique de l'espace des etats

We introduce a new method, which does not use replicas, from which we recover all the results of the replica symmetry-breaking solution of the Sherrington-Kirkpatrick model.

We propose and analyse a replica symmetric solution for random link travelling salesman problems. This gives reasonable analytical estimates for thermodynamic quantities such as the length of the shortest path Proposition et analyse d'une solution symetrique dans les repliques pour les problemes de voyageurs de commerce a liens independants. Estimations analytiques raisonnables pour les quantites thermodynamiques comme la longueur du chemin le plus court dans le cadre de l'etude des verres de spin