Douglas Abraham, Oxford University, England
Lecture 1: The interface between coexisting phases at equilibrium in the planar Ising model will be analysed using statistical mechanics, beginning with Gallavotti’s theorem. Some rather recent developments will be discussed and some unsolved (but potentially solvable) problems will be dscribed.
Lecture 2: We will discuss Fermi lattice gases and how the spectrum used in the previous lecture was obtained. The matrix elements used there, which we will also calculate (at least in outline), can be understood as an extension of the thinking in the Wick-Isserlis theorem. (Leon Isserlis (1881-1966) was a Russian-British statistician born in Kiev and a direct descendent of the eminent rabbi Moses Isserlis of Krakow.
Marek Biskup, University of California, Los Angeles, USA
Gradient models with convex, and non-convex, interactions
Gradient models are ubiquitous in statistical mechanics where they serve as models of surfaces and interfaces, crystal deformations, and fluctuation fields in spin systems. As far as mathematics is concerned, a rather complete theory exists in the case when the interactions are uniformly strictly convex. Unfortunately, this is hardly satisfied in many physical instances where such models arise. I will discuss specific examples of these models where the interaction is either just barely convex or non-convex at all, and where some progress can still be made.
Aernout van Enter, Groningen University, Netherlands
Ising models with long-range interactions, some recent results
We discuss some recent results about Ising models with polynomially decaying interactions.
1) We discuss how the phenomenon of entropic repulsion in the phase transition region can be used to show that Gibbs measures in one-dimensional (Dyson) models at low temperatures cannot be g-measures.
2) We discuss how one-dimensional models behave, when one adds external magnetic fields which decay at infinity, and in particular which rates of decay will still allow for a phase transition. We compare our results with
what happens in short-range models.
3) We discuss the (im)possibility of the existence of Dobrushin states in two-dimensional long-range models
Based on joint works with R. Bissacot, l. Coquille, E.Endo, B. Kimura, A.Le Ny, and W.M. Ruszel.
Peter Gacs, Boston University, USA
Reliable computation with cellular automata
Cellular automata are the most natural theoretical model in which to pose the problem of computing reliably. First since they possess parallelism, which is necessary to deal with noise of constant intensity. Second, due to their space-time homogeneity it is arguable that no structure has been taken for granted, other than the elementary geometrical properties of space and the "physics" or "chemistry" defining the transition function.
In the lectures, I will tell the story of constructing reliable cellular automata. I will start with the simplest model, a three-dimensional one that can be defined easily (though the proof that it works is nontrivial). Then I will outline a one-dimensional reliable cellular automaton. This is a complex hierarchical construction, and the exposition will focus on the main ideas.
The lectures will be similar in spirit to the ones here:
I attach a somewhat expanded version, not in slide form: it may be more readable for students.
3-dimensional reliable cellular automaton, based on Toom's rule:
2-dimensional reliable cellular automaton, using Toom's rule combined with hierarchy:
On 1-dimensional reliable cellular automata:
Discrete time: http://www.cs.bu.edu/faculty/gacs/papers/GacsReliableCA86.pdf
Continuous time, constant redundancy, self-organizing:
Daniel Stein, New York University – Courant, USA
Equilibrium statistical mechanics of short-range spin glasses
The aim of this minicourse is to introduce the subject of spin glasses, and more generally the statistical mechanics of quenched disorder,as a problem of general interest to mathematicians and physicists. Despite years of study, the physics and mathematics of quenched disorder remain poorly understood. While there are manyactive areas of investigation in this field, I will narrow the focus of this minicourse to our current level of understanding of the low-temperature equilibrium structure of realistic (i.e., finite-dimensional) spin glasses.
I will begin with a brief review of the basic features of spin glasses and what is known experimentally. I will then turn to the problem of understanding the nature of the spin glass phase --- if it exists. The central question to be addressed is the nature of broken symmetry in these systems. Parisi's replica symmetry breaking approach, now mostly verified for mean field spin glasses, attracted great excitement and interest as a novel and exotic form of symmetry breaking. But does it hold also for real spin glasses in finite dimensions? This has been a subject of intense controversy, and although the issues surrounding it have become more sharply defined in recent years, it remains an open question. I will explore this problem, introducing new mathematical constructs such as the metastate along the way.
1) K. Binder and A. P. Young, "Spin glasses: experimental facts, theoretical concepts, and open questions", Rev. Mod. Phys. 58, 801-976 (1986)
2) H.O. Georgii, "Gibbs Measures and Phase Transitions" (de Gruyter, 1988) (book)
3) M. Mezard, G. Parisi and M.A. Virasoro, "Spin Glass Theory and Beyond"
(World Scientific, 1987) (book containing all the classic papers on mean-field theory and the Parisi solution)
4) D.L. Stein and C.M. Newman, "Spin Glasses and Complexity" (Princeton, 2013)
5) R. Rammal, G. Toulouse and M.A. Virasoro, "Ultrametricity for Physicists", Rev. Mod. Phys. 58, 765-788 (1986)
Charles Radin, University of Texas at Austin, USA
Solids: ordered structures at high density
The lectures will discuss an interconnected series of topics:
i) densest packings (including aperiodic tilings)
ii) statistical mechanics and the law of large numbers
iii) the hard sphere model and quasicrystal models at high density
iv) nonstandard statistical mechanics for large constrained graphs, and for sand.
References C. Radin, Orbits of orbs: sphere packing meets Penrose tilings, Amer. Math. Monthly 111(2004), 137-149. C. Radin, A revolutionary material, Notices Amer. Math. Soc. 60(2013), 310-315. J. Ginibre, On some recent work of Dobrushin,Systemes a un nombre infini de degres de liberte, CNRS, Paris, 1969, pp. 163-175. D. Aristoff and C. Radin, First order phase transition in a model of quasicrystals, J. Phys. A: Math. Theor. 44(2011), 255001. C. Radin, Phases in large combinatorial systems, Ann. Inst. H. Poincare D (to appear), arXiv:1601.04787v2. Publications of Charles Radin - University of Texas at Austin http://www.ma.utexas.edu/users/radin/papers.html
Alexander Shen, LIRMM, Montpellier, France
Aperiodic tilings: fixed-point and other constructions
A classical result (Berger--Robinson theorem) says that there exists a tile set that allows tilings but only aperiodic ones. There are several proofs of this result: one may construct explicit self-similar tile set and check (by some case analysis) that it is self-similar (different versions of Robinson's proof); one may use the fact that 2^k\ne 3^l and provide a tiling that performs on each horizontal line a multiplication operation with factor $2$ or $1/3$ (Kari's proof), or use Kleene fixed-point theorem to get a self-similar tiling that computes its own rules (following old ideas of von Neumann and G\'acs). The latter approach does not provide a small number of tiles but is rather flexible and allows us to construct tile sets with additional properties including some robustness properties (every tiling with a sparse set of errors is close to an aperiodic one). We will discuss these approaches (following the audience's requests). [See https://arxiv.org/pdf/0910.2415.pdf and references within].
Siamak Taati, Toronto, Canada
Stable quasicrystal phases via cellular automata
I will present a finite-range lattice gas model that has "quasicrystal phases" at positive temperature. The construction is based on three ingredients from the theory of cellular automata (CA) and tilings: 1) a method of simulating a CA with another CA that is resilient against noise, due to Toom (1980), Gacs and Reif (1988), 2) the existence of aperiodic sets of Wang tiles that are deterministic in one direction (e.g., based on Ammann's golden tiles,~1980), 3) the observation going back to Domany and Kinzel (1984) and developed by Goldstein, Kuik, Lebowitz, and Maes (1989) that the space-time diagrams of positive-rate probabilistic CA are Gibbs measures. I will provide some details about the construction, discuss the properties and shortcomings of the model, and pose several open questions.
Jan Dereziński, University of Warsaw, Poland
Homogeneous Schroedinger operators on halfline
I will discuss the theory of Schr\"odinger operators with 1/x^2 potentials. This is a class of objects with surprisingly rich mathematical phenomenology, which should be close to physicists' hearts: the "running coupling constant" flows under the action of the "renormalization group", there are two "phase transitions", attractive and repulsive fixed points, limit cycles, breakdown of conformal symmetry, etc. I will discuss both the self-adjoint and non-self-adjoint cases. The latter have quite curious properties and I am looking for their physical applications.
Henna Koivusalo, University of Vienna, Austria
Jacek Miękisz, IMPAN and MIMUW, Warsaw, Poland
From Hilbert to Shechtman - a brief history of quasicrystals
The first part of the presentation will be an introduction to the mathematics of quasicrystals as seen from the theoretical physicist point of view. We will discuss simple example of optimization problems in kissing numbers, packing spheres, non-periodic tilings, and classical-lattice gas models of statistical mechanics. We will also formulate the main open problem - the existence of non-periodic Gibbs measures for finite-range hamiltonians.
In the second part of the talk we will discuss various connect ions between ergodic theory, substitution dynamics, dynamical systems of finite type, and classical lattice gas-models.
Wioletta Ruszel, Technical University Delft, Netherlands
Sandpile models, random interfaces and scaling limits
In this talk we will introduce a special probabilistic cellular automaton, namely the (divisible) sandpile model and discuss connections with random interfaces and their (continuous space) scaling limits.
The divisible sandpile model (DSM) is the continuous height counterpart of the Abelian sandpile which is a toy model introduced by Bak, Tang and Wiesenfeld in 1987 displaying self-organized criticality. Self-organized critical models are in some sense driving themselves into a critical state without fine-tuning of any external parameters such as the temperature. DSM were used in connections with internal diffusion limited aggregation growth models. In the DSM setting one is starting with a random continuous initial height configuration and if the height exceeds 1, keeping mass 1 and toppling the excess to some neighbours with equal probability. It turns out that under some conditions the final configuration will be the all 1 configuration and that the odometer which records the amount of mass emitted during the stabilization is defining a random interface configuration. We will discuss this construction and under which conditions the random interface is of Gaussian, alpha-stable or fractional type and determine their continuous scaling limit.
This is joint work with: Alessandra Cipriani (TU Delft), Rajat Hazra (ISI Kolkata), and Leandro Chiarini (TU Delft).
Cristian Spitoni, Utrecht University, Netherlands
Nucleation for Probabilistic Cellular Automata with self-interaction
Metastable states are very common in nature and are typical of systems close to afirst order phase transition. Classical examples are the supersaturated vapor and the magnetic hysteresis. The full mathematical description of metastability is quite recent and still incomplete. In this framework, Probabilistic Cellular Automata (PCA) pose challenging problems and show unexpected behaviours. Cellular Automata (CA) are discrete-time dynamical systems on a spatially extended discrete space.
PCA have been introduced as a stochastic generalization of CA. In this lecture we shall focus on the study of metastability for a class of reversible PCA, trying to illustrate the extremely rich phenomenology encountered when the parameters of the PCA are tuned. In particular, the dependence of the metastability scenario (e.g. the nucleation process of the stable phase) on the self- interaction parameter will be discussed. In the absence of self-interaction an intermediate metastable state shows up (two flip-flopping chessboard configurations). The role played by the intermediate state changes as the self-interaction ? is varied. The talk will review rigorous results and heuristics for open problems.
Jan Wehr, University of Arizona, Tucson, USA
Uniqueness of zero-temperature metastates in random Ising ferromagnets
We study an Ising model with random ferromagnetic coupling constants. At zero temperature this model has at least two ground states, in which all the spins take the same value (either +1 or -1). The question about possible existence of other ground states is open---even in two dimensions where there seems to be a universal agreement that the constant ground states are the only ones. Zero temperature *metastates* are probability measures on ground states which depend covariantly on the coupling realization. Our main result is that such metastates are supported only on constant ground states in any dimension. This offers a new perspective on the ground state problem and suggests further strategies to resolve it. Relations with first-passage models in two dimensions and minimal-surface models in higher dimensions will be discussed. The work has been done jointly with Aramian Wasielak.