Tag - Statistical mechanics

Huy Tuan Pham: Structures in Random Graphs: New Connections

28th July, 2023

In 1947, Paul Erdos famously gave a construction of graphs with no large cliques and independent sets via the probabilistic method. Since then, Erdos’ ingenious probabilistic insight and the emergence of structures in random graphs have motivated fundamental developments in combinatorics and probability. In this talk, I highlight several recent results in random graph studies with interesting connections to additive combinatorics, theoretical computer science and probability.

In joint work with David Conlon, Jacob Fox and Liana Yepremyan, we study structures in random Cayley graphs of general groups. Given a fixed finite group, random Cayley graphs are constructed by choosing the generating set at random. These graphs thus reflect interesting symmetries and properties of the group, at the cost of inducing complex dependencies. We will discuss results on clique and independence numbers in random Cayley graphs of general groups, as well as progress towards a conjecture of Alon on Cayley graphs with small clique and independence number. These questions are naturally connected with some fundamental problems in additive combinatorics, where we surprisingly discover that in some of them the group structure is superfluous.

Certain important aspects in the study of structures in random graphs can be phrased in terms of thresholds. In joint work with Jinyoung Park, building on insights in our resolution of Talagrand’s selector process conjecture, we prove the Kahn-Kalai conjecture, that thresholds of general monotone properties are closely predicted by expectation obstructions. The Kahn-Kalai conjecture is a beautiful milestone towards the understanding of emergence of general structures, and yet to complete the quest, it remains to study these expectation obstructions. This latter task can prove to be highly challenging in several cases and bring in interesting connections. As an illustration, I will discuss joint work with Vishesh Jain that determines the threshold for edge-colorings of complete bipartite graphs by exploiting connections to the Lovasz Local Lemma and local uniformity properties, which have played an important role in sampling solutions to constraint satisfaction problems.

Francesco Buscemi: Various types of divisibility and the role they play in statistical mechanics

23rd May, 2023

In this talk, I will argue that besides the conventional divisibility property (that is, the semigroup property), other types of divisibilities arise naturally in the context of open systems' dynamics. The common thread connecting them all is a general idea of 'inferential locality', which is necessary when discussing the physics of open systems. As concrete examples, I will focus on the problems of system-bath divisibility and prediction-retrodiction divisibility, and explain their role within the conceptual foundations of statistical mechanics.

Amol Aggarwal: Universality results in random tiling models II

7th April, 2023

Random tilings of large domains have long been objects of fascination for mathematicians and physicists. What one witnesses for these systems is that the shapes of their domains have a significant impact on the local geometry of the tiling. Indeed, these shapes influence the probabilities of (microscopic) tiling patterns well inside the domain; can induce phase transitions (arctic curves) at facet edges; and can create "turning points" where the limiting behaviour of the model is discontinuous. At least for certain special shapes, there now exist powerful algebraic frameworks providing exact formulas, which can be analysed to precisely identify the limiting statistics that appear in each of these situations. This gives rise to a universality prediction, that the same limiting statistical behaviours should also arise in random tilings not only of those particular domains, but also of far more general ones. Over the past several years, there has been progress in establishing such universality statements for random tilings of the triangular lattice, or lozenge tilings. In these talks, we review some of these developments, and outline how they are obtained, by intertwining the above mentioned algebraic frameworks with analytic and probabilistic ones.

Amol Aggarwal: Universality results in random tiling models I

7th April, 2023

Random tilings of large domains have long been objects of fascination for mathematicians and physicists. What one witnesses for these systems is that the shapes of their domains have a significant impact on the local geometry of the tiling. Indeed, these shapes influence the probabilities of (microscopic) tiling patterns well inside the domain; can induce phase transitions (arctic curves) at facet edges; and can create "turning points" where the limiting behaviour of the model is discontinuous. At least for certain special shapes, there now exist powerful algebraic frameworks providing exact formulas, which can be analysed to precisely identify the limiting statistics that appear in each of these situations. This gives rise to a universality prediction, that the same limiting statistical behaviours should also arise in random tilings not only of those particular domains, but also of far more general ones. Over the past several years, there has been progress in establishing such universality statements for random tilings of the triangular lattice, or lozenge tilings. In these talks, we review some of these developments, and outline how they are obtained, by intertwining the above mentioned algebraic frameworks with analytic and probabilistic ones.

Sky Yang Cao: Recent Results on Finite Group Lattice Gauge Theories

20th March, 2023

The rigorous study of spin systems such as the Ising model is currently one of the most active research areas in probability theory. In this talk, I will introduce one particular class of such models, known as lattice gauge theories (LGTs), and go over its origins, motivations, and then some recent results. Along the way, I will also try to highlight some of the key differences between LGTs and the usual spin systems. The general theme is that LGTs are spin systems with topological considerations.

Vladimir Bazhanov: Quantum geometry of 3-dimensional lattices

26th October, 2021

In this lecture I will explain a relationship between incidence theorems in elementary geometry and the theory of integrable systems, both classical and quantum. We will study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices, lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analogue of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.