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Meeting at "Il Palazzone", Cortona (Italy), June 27 - July 2, 2005

Promoted by INdAM

» Program:

(Tentatively, one full hour should also include a 5-10 minutes discussing time)

 Monday, June 27 Tuesday, June 28 Wednesday, June 29 Thursday, June 30 Friday, July 1 Saturday, July 2 9.00 - 10.00 Guerra Aizenman Guerra Newman Stein Talagrand 10.00 - 11.00 Aizenman Guerra Aizenman Stein Talagrand Nishimori 11.00 - 11.30 coffee break coffee break coffee break coffee break coffee break coffee break 11.30 - 12.30 Derrida Parisi Newman Talagrand Ben Arous Bovier 12.30 - 14.30 lunch lunch lunch lunch lunch lunch 14.30 - 15.30 Olivieri Franz Giardinà Guionnet De Sanctis 15.30 - 16.30 Contucci Marinari Toninelli Isopi 16.30 - 17.00 coffee break coffee break coffee break coffee break 17.00 - 18.00 Shcherbina Fröhlich Lebowitz Panchenko

On Wednesday, June 29, from 6.15 to 7 pm Joel Lebowitz will chair an Informal Session on Human Rights

Lectures
:

Michael Aizenman

Title:
Abstract:

Francesco Guerra

Title:
Overlap locking in mean field spin glasses

Abstract:
In the frame of interpolation methods, we show how comparison systems are able to lock their overlaps to those of a mean field spin glass.
Locking power is expressed in terms of overlap fluctuations. We give as examples the case of the annealed approximation, the replica
symmetric approximation, and the Parisi representation. A byproduct of our treatment is a simple proof that replica symmetry holds up to the Almeida-Thouless line.

Chuck Newman & Dan Stein

Title:
Short-Range Spin Glasses

Abstract:
We review results about short-range spin glasses at low and zero temperature.  We will focus on
what is presently known, and what we believe might be provable in the near future.  Our focus is on low-temperature
equilibrium properties of short-range spin glasses, including: numbers of pure and ground states in various dimensions;
the nature of excitations and interfaces and their connection to pure state structure; the use of metastates to analyze
the organization of the broken-symmetric structure of the low-temperature phase.

Michel Talagrand

Title:
Large Deviations, Guerra and A.S.S. schemes and the Parisi hypothesis

Abstract:
We consider the problem of computing $$\lim_{N \rightarrow \infty} \frac{1}{Na} E \log Z_N$$ when $a$ is a real number and $Z_N$ is the partition function of the SK model. We review in this setting some of the recent ideas that seem fruitful.

Seminars:

Gerard Ben Arous

Title:
The Arcsine law for stable subordinators as a universal mechanism of Aging for spinglass dynamics?

Abstract:

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Anton Bovier

Title:
The local REM conjcture and beyond

Abstract:
Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete
spin space should in most circumstances be the same as in the random energy model. Here we give necessary
conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin
glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if
energy levels that grow moderately with the volume of the system are considered. Finally, we show that the conjecture
fails if levels near the extremes are considered, and show that a complex and rich structure emerges in this case in the
example of Derrida's generalized random energy models.

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Pierluigi Contucci

Title:
The Ghirlanda-Guerra Identities

Abstract:
We will show a proof of the Ghirlanda-Guerra identities which only requires that the variance of the Hamiltonian grows like the volume (thermodynamic stability). Our result is expressed in terms of the model's covariance and applies to all known spin glass models.

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Bernard Derrida

Title:
Noisy traveling waves in evolution models and spin glasses

Abstract:
Models of evolution in presence of selection can be formulated as noisy traveling waves. This talk will review
some recent results obtained on the effect of noise on the velocity and of the diffusion constant of these noisy
traveling waves. The exact solution of a special case reveals statistical properties of trees which are
quantitatively identical to those of spin glasses, as predicted by the Parisi broken symmetry of replica.

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Luca De Sanctis

Title:
Overlap Structure approach to various spin glass models

Abstract:
We show how Random (Multi-) Overlap Structures can be used to approach various spin glass models, in particular
diluted systems and optimization problems. We illustrate some advantages and disadvantages of possible Ansatzs
that can be formulated in such context

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Silvio Franz

Title:
A field-theoretical approach to the spin glass transition

Abstract:
We study spin glasses with Kac type interaction potential for small but finite inverse interaction range $\gamma$. Using the
theoretical setup of coupled replicas, through the replica method we argue that the probability of overlap profiles can be
expressed for small $\gamma$ through a large-deviation functional. This result is supported by rigorous arguments, showing
that the large-deviation functional provides at least upper bounds for the probability. Finally we analyze the rate function, in the
vicinity of the critical point $T_c=1$ of mean field theory, and we study the free energy cost of overlap interfaces, assuming the
validity of a gradient expansion for the rate functional.

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Jürg Fröhlich

Title:
Abstract:

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Cristian Giardina'

Title:
On the (link) overlap distribution on the Edwards-Anderson model.

Abstract:
In the first part of the talk, we will give a rigorous formulation of the Stochastic Stability property for a large class of Gaussian spin
glass models. From this we will obtain some well-known identities among overlaps moments, which are included in the Ghirlanda-Guerra
identities. Then we specialize to Edward-Anderson model for which we present a numerical study of the overlap factorization properties.

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Alice Guionnet

Title:
Dynamics for spherical SK dynamics

Abstract:
We consider the Langevin dynamics for p-spins model of Sherrington-Kirkpatrick model. We prove convergence of the empirical distribution of
the particles. The limit is described by a coupled integro-differential system involving the dynamical covariance and the response function.
We discuss the long time asymptotics of the solution in the high temperature regime and the FDT regime, following works of Cugliandolo,
Kurchan et al.

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Marco Isopi

Title:
Coarsening on a one-dimensional lattice: the continuum limit

Abstract:
Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating)
random walks representing the evolving boundaries between regions of constant color and by backward (in time)
coalescing random walks corresponding to color genealogies. Asympotics for large time and space on the lattice are
described via a continuum space-time voter model whose boundary motion is expressed by the Brownian web (BW) of coalescing
forward Brownian motions. I will describe how small noise in the voter model, corresponding to the nucleation of randomly
colored regions, can be treated in the continuum limit.

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Joel Lebowitz

Title:
Fluctuations, Large Deviations, and the Custom Construction of Point Processes

Abstract:
I will review some aspects of fluctuations in equilibrium and nonequilibrium systems with emphasis on their relation to large
deviations and entropy. I will then describe some recent work on the construction of point processes with specified low order correlations.
These specify, among other things, the variance of the fluctuations of local quantities.

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Enzo Marinari

Title:
Low T scaling behavior of 2D disordered and frustrated models.

Abstract:
The ground state and low T behavior of two-dimensional spin systems with discrete binary couplings are subtle. I present an analysis based
on exact computations of finite volume partition functions. I first discuss the fully frustrated model without disorder, and then introduce
disorder by changing random links (spin glass) or by unfrustrating random plaquettes (plaquette disorder). In both cases
the introduction of disorder changes the properties of the T=0 critical point.

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Hidetoshi Nishimori

Title:
Duality in spin glasses

Abstract:
I prove the existence of duality relations in finite-dimensional spin glass models under replica formalism.
The models are shown to be self-dual under certain conditions. Assuming analyticity of duality transformation, I derive
relations between transition points, which determine the critical point if it is unique. If one accepts the zero-replica limit
as a legitimate replacement of average over quenched randomness, our method gives the (possibly exact) location
of the multicritical point in the phase diagram of quenched system. The results are in impressive agreement with
numerical investigations in all available cases.

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Enzo Olivieri

Title:
Disordered systems close to criticality.

Abstract
:
We consider disordered lattice spin systems. In particular we analyze small random perturbations of strong mixing systems
giving rise to the Griffiths' phase. We use a graded cluster expansion with a minimal length depending on the thermodynamic
parameters. On the lowest length scale, contrary to the usual high temperature or large magnetic field perturbative theories
we use a "scale-adapted cluster expansion " based on a suitable finite size mixing condition. We rely on a general theory
already applied to the problem of proving weak Gibbsianity and convergence of the iterates for a class of renormalization
group maps.

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Dmitry Panchenko

Title:
On the generalized Sherrington-Kirkpatrick model.

Abstract:
I will present the extension of the Talagrand's proof of the Parisi formula to a general class of the SK-type models with spins
distributed according to an arbitrary probability measure on a bounded subset of the real line. I will describe the analogue
of the Parisi formula, the replica symmetric region and talk about some properties of the Parisi functional.

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Giorgio Parisi

Title:
The art of computing complexity

Abstract:
In this talk I will review some of the recent results in computing the exponentially large nmber of metastable states (often identified with
the solutions of the TAP equations) in the infinite range limit (SK model) for spin glasses and on the Bethe lattice.

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Mariya Shcherbina

Title:
Cavity methods in the dynamics of neurons systems

Abstract:
The dynamics of systems of $N$ neurons governed by the system of liner or nonlinear differential equations with random
coefficients will be discussed. The statistical distribution (as $N\to\infty$) of neurons on the real line at any fixed time $t$
is studied using the cavity method.

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Fabio Lucio Toninelli

Title:
Random copolymers at a selective interface: phase diagram and some path properties

Abstract:
I will consider a model of a random heterogeneous polymer in the proximity of an interface separating two
selective solvents. This model exhibits a localization/delocalization transition. A positive value of the free
energy corresponds to the localized regime and strong results on the polymer path behavior are known in
this case. I will focus on the interior of the delocalized phase, which is characterized by the free energy equal to zero,
and I will show in particular that in this regime there are O(log N) monomers in the unfavorable solvent
(N is the length of the polymer). The previously known result was o(N). Another result concerns the weak coupling
limit: we show the universal nature of this limit, previously considered only for binary disorder. Although the
physics of the model is quite far from that of spin glasses, I will show that the mathematical tools which
are essential in the solution of the SK model (i.e., interpolation and concentration of measure) play an essential
role also in this case.

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