November 9, 2012
Location: Janskerkhof 15a (Utrecht), room 101
Stochastic representations of quantum Gibbs states I: General setup and examples
I shall talk about stochastic representations which are based on the path integral approach.
Introduction of an additional dimension (imaginary time) enables a reinterpretation in terms of classical probabilities. Although a relation to the original quantum model sets up the stage for pending questions, the emerging classical models are frequently interesting even in their own right. For instance, such procedure naturally leads to space-time generalizations of classical models of percolation, including a space-time version of Erdős–Rényi random graph.
Furthermore, the path integral approach gives an alternative way to think about stochastic geometry of classical models of statistical mechanics, e.g. FK and random current representations of ferromagnetic Ising models. Both happen to have natural quantum generalizations.
I shall assume an at most superficial knowledge of quantum mechanics, and try to keep the exposition self-contained and suitable for a probabilistic audience. However, familiarity with classical models, such as Ising or percolation, would be helpful.
Two main examples I plan to discuss during this talk are: Ising model in transverse field and quantum Heisenberg model.
Large deviations for the cluster size distributions in a classical interacting many-particle system with Lennard-Jones potential
We study a classical many-body system with pair-interaction given by a stable Lennard–Jones potential. This interaction has an attractive term, which induces the formation of clusters of the particles. For fixed inverse temperature β ∈ (0,∞) and fixed particle density ρ ∈ (0,∞), we derive a large-deviation principle for the distribution of the cluster sizes in the thermodynamic limit. Afterwards, we show that the rate function Gamma-converges, in the low-temperature dilute limit β → ∞ and ρ ↓ 0 such that -β-1 log ρ → ν ∈ (0,∞), towards some explicit rate function. This function has precisely one minimising cluster size configuration, which implies a law of large numbers for the cluster sizes in this decoupled limit. This is joint work with S. Jansen and B. Metzger. The limiting rate function appeared in earlier work with A. Collevecchio, P. Mörters and N. Sidorova, when the two limits (the thermodynamic and low-temperature dilute limit) were coupled with each other.