Wednesday, March 24, 2021

11:00 a.m. (sharp)

PhD Candidate: Larissa Richards

Supervisor: Ilia Binder

Thesis title: Convergence rates of random discrete model curves approaching SLE curves in the scaling limit

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Recently, A. Kempannien and S. Smirnov provided a framework for showing convergence of discrete

model interfaces to the corresponding SLE curves. They show that given a uniform bound on specific

crossing probabilities one can deduce that the interface has subsequential scaling limits that can be

described almost surely by Löwner evolutions. This leads to the natural question to investigate the

rate of convergence to the corresponding SLE curves. F. Johansson Viklund has developed a framework

for obtaining a power-law convergence rate to an SLE curve from a power-law convergence rate for the

driving function provided some additional geometric information along with an estimate on the growth

of the derivative of the SLE map. This framework is applied to the case of the loop-erased random

walk. In this thesis, we show that if your interface satisfies the uniform annulus condition proposed by

Kempannien and Smirnov then one can deduce the geometric information required to apply Viklund’s

framework. As an application, we apply the framework to the critical percolation interface. The first

step in this direction for critical percolation was done by I. Binder, L. Chayes and H.K. Lei where they

proved that the convergence rate of the Cardy-Smirnov observable is polynomial in the size of the lattice.

It relies on a careful analysis of the boundary behaviour of conformal maps and their discrete analytic

approximations as well as a Percolation construction of the Harris systems. Further, we exploit the

toolbox developed by D. Chelkak for discrete complex analysis on isoradial graphs to show polynomial

rate of convergence for the discrete martingale observables for harmonic explorer and the FK Ising

model to the corresponding continuum objects. Then, we apply the framework developed above to gain

a polynomial convergence rate for the corresponding curves.

A copy of the thesis can be found here: LarissaRichardsThesis

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