Schedule
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Wednesday, July 12, 2023 ›
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09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
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›9:30 (1h30)
Ioan Manolescu - Exploring the critical phase of 2D FK-percolation, Part 2
In these lectures we will study FK-percolation (also called the random-cluster model) on the square lattice Z^2. Our focus will be on recent results describing the phase transition, namely: - its sharpness; - its continuity/discontinuity depending on the cluster parameter q; - the large-scale rotational invariance of the critical models. These results have been obtained through a wide variety of techniques; of particular interest will be the relationship between FK-percolation and the six-vertex model, and the treatment of the latter using the transfer matrix formalism.
› Amphi 15
›14:30 (45min)
Filippo Colomo - Arctic curves of the four-vertex model
We consider the four-vertex model with a particular choice of fixed boundary conditions. In the scaling limit, the model exhibits the limit shape phenomenon, with the emergence of an arctic curve separating a central disordered region from six frozen "corners" of ferroelectric or anti ferroelectric type. We determine the analytic expression of the arctic curve. The derivation is based on the exact evaluation of suitable correlation functions, discriminating spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to Hahn polynomials. As a by-product, we also deduce that the arctic curve's fluctuations are governed by the Tracy-Widom distribution. Joint work with I.N. Burenev, A. Maroncelli, and A.G. Pronko
› Amphi 15
›15:15 (45min)
David Keating - Double dimers, coupled tilings, and LLT polynomials
We study $k$-tuples of tilings of the Aztec diamond in which we assign a weight to each tuple depending on the number of 'interactions' between the tilings. There are several ways to view these tilings: as a double dimer model with an interaction between the dimers, as an LLT process generalizing the usual Schur process, or as a certain integrable vertex model. We will define the coupled tilings relating them to each of these settings. Then we will describe a generalization of the domino shuffling algorithm to the coupled tilings.
› Amphi 15
›16:30 (45min)
Richard Kenyon - Dimers and 3-webs
Associated to a GL_n-local system on a graph on a surface is a combinatorial formula enumerating traces of n-webs in the graph, in terms of a determinant of an associated matrix, the Kasteleyn matrix. We explain this formula and show how, for SL_3, it can be used to enumerate webs of given topological types, in the case of graphs in simple surfaces. This is based on joint work with Dan Douglas and Haolin Shi.
› Amphi 15
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Amphi 15
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