3.3 Equilibrium Statistical Mechanics

Organizer C.M. Newman (New York)

3.3.1 The Scaling Limit of (Near-)Critical 2d Percolation

Federico Camia

Vrije Universiteit Amsterdam fede@few.vu.nl

The introduction of the random fractal curves described by Schramm-Loewner Evolutions (SLEs) has greatly deepened our understanding of the large-scale structure and fractal properties of certain two-dimensional lattice models whose continuum scaling limit is known or conjectured to be conformally invariant.

A particularly illuminating example is that of critical percolation, where the connection with SLE can be made rigorous and can be exploited to describe the scaling limit of the model in terms of the collection of all its interfaces.

In turn, this description can be used to analyze the scaling limit of nearcritical percolation and related models. The emerging picture suggests that the conformal invariance typical of critical models is replaced by a more general type of invariance under conformal transformation.

In this talk I will describe joint work with C.M. Newman and with L.R. Fontes and C.M. Newman on the scaling limit of critical and near-critical percolation.

3.3.2 Short-Range Spin Glasses in a Magnetic Field

Daniel Stein+

NYU daniel.stein@nyu.edu

The thermodynamic behavior of short-range spin glasses in small magnetic fields and at low temperatures remains an open problem. Results elucidating the consequences of applying a magnetic field can provide a powerful tool in understanding the low-temperature behavior of realistic spin glasses, in particular the nature of their broken symmetry, phase diagram, and other fundamental thermodynamic information. I will discuss new techniques that have recently been developed to treat this problem, and present some preliminary results that clarify the effects of a magnetic field on short-range spin glasses.

In collaboration with C.M. Newman, Courant Institute, New York University. +Research supported by the U.S. National Science Foundation.