When an abstract planar graph is endowed with a random walk, various planar embeddings of this graph can be constructed. Classical examples are of Tutte's harmonic embeddings and Brooks-Smith-Stone-Tutte square tilings; they are generalized by the so-called T-graphs introduced by Kenyon and Sheffield. Given a sequence of such embeddings discretizing a given planar domain, it is natural to ask how the scaling limit of the random walk can be described in terms of the embeddings. In this talk we will address this question in the context of T-graphs. The key role in our approach is played by t-embeddings of bipartite graphs and the regularity theory of discrete holomorphic functions on them developed by Chelkak, Laslier and Russkikh. Based on a joint work with Chelkak, Laslier and Russkikh.
In the first part of the talk, I will introduce some basic ideas of SLE theory, and discuss one of our results in the one SLE curve case, concerning the law of the SLE tip at a fixed time. In the second part of the talk, I will focus on the multiple SLE curves, and I will describe a new toolbox that connects Multiple Schramm-Loewner Evolutions and Random Matrix Theory. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. I will describe the connection with Random Matrix Theory via a first application of our method. I will also discuss some open problems that emerge both in the one SLE curve and Multiple SLE curves picture. The first part is a joint work with O. Butkovsky and Y. Yuan, and the second part is a joint work with A. Campbell and K. Luh.
We give an overview of various approaches to quantum field theory in general and an algebraic framework specific to two-dimensional CFT, (full) vertex operator algebras. One can define correlation functions in a full vertex algebra under some regularity conditions. We show that, under unitarity, they satisfy the Osterwalder-Schrader axioms. We sketch how the Wightman fields are recovered. (joint with Maria Stella Adamo, Luca Giorgetti and Yuto Moriwaki)
In this talk we describe the work (arXiv:2501.19014) in collaboration with B. Estienne where we demonstrate a purely mathematical and geometric construction which recovers some results of Cardy and Calabrese on the so-called entanglement entropy. We start by explaining briefly the Segal picture of CFT (and QFT), and its natural relation to path-integral formulation. Then we introduce the entanglement entropy, and relate it to conical surfaces via path-integral and the "replica trick". Then we say briefly about geometry of conical singularities. Finally we formulate the main result of the work: we define CFT "partition functions" on surfaces with conical singularities, using a "Hadamard renormalization'' of the Polyakov anomaly integral. Then for a branched cover $f:\Sigma_d→\Sigma$ of degree $d$, the ratio $Z(\Sigma_d,f^*g)/Z(\Sigma,g)^d$ of partition functions transforms under conformal changes of g like a correlation function of CFT primary operators of specific conformal weights.
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