ICMP 2003 > Sessions > Equilibrium statistical mechanics

# Equilibrium statistical mechanics

Session organized by D. Brydges (Vancouver), E. Olivieri (Rome).

## Talks

 L. Chayes UCLA F. Guerra Rome J. Imbrie Charlottesville D. Ioffe Haifa H. Knoerrer Zürich G. F. Lawler Cornell

Lincoln Chayes
Department of Mathematics
UCLA

Thresholds for surface formation in equilibrium: mesoscopic and macroscopic phenomena

The topic to be presented concerns the physics of surface formation in the context of equilibrium statistical mechanics. Explicitly, for systems at phase coexistence, we shall underscore the existence of sharp thresholds (in the scaling sense) above which there are droplets and below which the system is homogenous. Depending on the setup -- i.e. the statistical ensemble -- the threshold droplet size can be macroscopic (and essentially trivial) or can occur at scales that diverge with system size but occupy a vanishing fraction of the system. The latter cases are, evidently, examples of classical mesoscopic phenomena. Finally, examples will be discussed which are of a mixed nature: macroscopic threshold droplets which subsume only a small fraction of the system. If time permits, a few applications will be touched upon, in particular the Gibbs-Thompson effect and the threshold scaling for the onset of freezing point depression.

Francesco Guerra
Dipartimento di Fisica
University of Rome "La Sapienza"

Rigorous results for mean field spin glass models

We will give a short comprehensive review about recent rigorous results in the study of mean field spin glass models. The results include the infinite volume limit, sum rules for the free energy, bounds coming from the broken replica symmetry Ansatz, the high temperature behavior, properties of the ground state energy. It will be also shown how these techniques extend to diluted models, the K-SAT problem, and the assigment problem. Most of the results have been obtained in collaboration with Fabio Lucio Toninelli.

John Z. Imbrie
Department of Mathematics
University of Virginia
Charlottesville
VA 22904-4137
USA

Branched polymers and dimensional reduction

I will describe an exact relation between self-avoiding branched polymers in D+2 continuum dimensions and the hard-core continuum gas at negative activity in D dimensions (joint work with David Brydges, [BI01]). Our result explains why the critical behavior of branched polymers should be the same as that of the i\varphi3 (or Yang-Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas [PS81]). I will review conjectures and results on critical exponents for D+2 = 2,3,4 and show that they are corollaries of our result. We have also derived the form of the massive scaling limit of the two-point function for branched polymers in three dimensions \cite[BI03]. I will also discuss directed branched polymers in D+1 dimensions, and show that they, too, are related to the hard-core gas in D dimensions. \end{abstract}

[BI01] D.C. Brydges and J.Z. Imbrie, "Branched polymers and dimensional reduction", preprint [arXiv:math-ph/0107005].
[BI03] D.C. Brydges and J.Z. Imbrie, "Dimensional reduction formulas for branched polymer correlation functions", J. Statist. Phys. 110, 503-518, 2003 [arXiv:math-ph/0203055].
[PS81] G. Parisi and N. Sourlas, "Critical behavior of branched polymers and the Lee-Yang edge singularity", Phys. Rev. Lett. 46, 871-874, 1981.

Dmitry Ioffe
Israel Institute of Technology
Technion

Random walk representation and Ornstein-Zernike theory of fluctuations

Rigorous approach to the Ornstein-Zernike theory of fluctuations is based on a random walk/random line type representation of semi-invariants (high temperature models) and, respectively, phase boundaries (low temperature two-dimensional models) and on renormalization procedures which set up the stage for a probabilistic analysis along the lines of the thermodynamic formalism of one-dimensional systems associated to countable Markov shifts.

Recent results include sharp asymptotics of correlation functions for sub-critical percolation models/high-temperature finite range Ising models in any dimension and an invariance principle for low temperature two-dimensional interfaces, in particular for phase separation lines in nearest neighbour Ising model in the whole of the phase transition region.

Horst Knoerrer
epartment of Mathematics
ETH - Zürich

Construction of a 2-d Fermi liquid: the strategy

This is one of two self-contained talks at ICMP 2003 concerning the proof (of Joel Feldman, Eugene Trubowitz and H. K.) that the temperature zero renormalized perturbation expansions of a class of interacting many-fermion models in two space dimensions have nonzero radius of convergence. The models have "asymmetric" Fermi surfaces and short range interactions. One consequence of the convergence is the existence of a discontinuity in the particle number density at the Fermi surface. The detailed proof is now available on the net at http://www.math.ubc.ca/~feldman/. This talk will highlight the strategy of the construction. The other, which is in the session on Condensed Matter Physics, will concentrate on the statement of results.

Gregory F. Lawler
Department of Mathematics
Cornell University

Restriction property and the Schramm-Loewner evolution

There are two important classes of conformally invariant measures on clusters in two dimensions --- restriction measures and Schramm-Loewner evolution (SLE). These both arise as limits of models in statistical physics in two dimensions. I will discuss the relationship between them and how "universality" properties can be used to conjecture and/or prove scaling limits of discrete systems. This is joint work with Oded Schramm and Wendelin Werner.




## Posters

A.C.D. Van Enter, S.B. Sholsman
RUG, Netherlands

First-order transitions in vector models and lattice gauge models with continuous symmetries

A.N. Ivanova, S.I. Kuchanov, L.I. Manevitch
Inst. Probl. Chem. Phys. RAS, Moscow, Russia; Inst. Appl. Math. RAS, Mocow, Russia and Inst. Chem. Phys. RAS, Moscow, Russia

Bifurcation analysis for the construction of a phase diagram of heteropolymer liquids

P. Martin, J. Piasecki
ITP, SFIT, Lausanne, Switzerland and Inst. Th. Phys., Univ. Warsaw, Poland

Quantum Mayer graphs and self-consistent equation for an interacting Bose gas

P.L. Ferrari, H. Spohn
ZM, TU, Munchen

Shape fluctuations of a faceted crystal

W.L. Spitzer, S. Starr
UC Davis and Princeton, USA

Improved bounds on the spectral gap above frustration free ground states of quantum spin chains

Finite size corrections for Coulomb systems at the Debye-Huckel limit

V.A. Zagrebnov
Univ. de la Mediterranée, CPT, Marseille, France

Bose-Einstein condensation for homogeneous intercating systems with a one-particle spectral gap

S.L. Starr
Princeton, USA

Ferromagnetic ordering of energy levels

K.R. Ito
Setsunen U.

Absence of phase transitions in two-dimensional O(N) spin models and Anderson localization

J-N. Aqua, M.E. Fischer
Maryland

Critical phenomena in ionic fluids: the spherical model

C. Tutshcka
Univ. Carlos III de Madrid, Spain

Partitions in the theory of density functionals