Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.707786 
Title:  Exploring random geometry with the Gaussian free field  
Author:  Jackson, Henry Richard 
ORCID:
0000000169953051
ISNI:
0000 0004 6056 9987


Awarding Body:  University of Cambridge  
Current Institution:  University of Cambridge  
Date of Award:  2016  
Availability of Full Text: 


Abstract:  
This thesis studies the geometry of objects from 2dimensional statistical physics in the continuum. Chapter 1 is an introduction to SchrammLoewner evolutions (SLE). SLEs are the canonical family of nonselfintersecting, conformally invariant random curves with a domainMarkov property. The family is indexed by a parameter, usually denoted by κ, which controls the regularity of the curve. We give the definition of the SLEκ process, and summarise the proofs of some of its properties. We give particular attention to the RohdeSchramm theorem which, in broad terms, tells us that an SLEκ is a curve. In Chapter 2 we introduce the Gaussian free field (GFF), a conformally invariant random surface with a domainMarkov property. We explain how to couple the GFF and an SLEκ process, in particular how a GFF can be unzipped along a reverse SLEκ to produce another GFF. We also look at how the GFF is used to define Liouville quantum gravity (LQG) surfaces, and how thick points of the GFF relate to the quantum gravity measure. Chapter 3 introduces a diffusion on LQG surfaces, the Liouville Brownian motion (LBM). The main goal of the chapter is to complete an estimate given by N. Berestycki, which gives an upper bound for the Hausdor dimension of times that a γLBM spends in αthick points for γ, α ∈ [0, 2). We prove the corresponding, tight, lower bound. In Chapter 4 we give a new proof of the RohdeSchramm theorem (which tells us that an SLEκ is a curve), which is valid for all values of κ except κ = 8. Our proof uses the coupling of the reverse SLEκ with the free boundary GFF to bound the derivative of the inverse of the Loewner flow close to the origin. Our knowledge of the structure of the GFF lets us find bounds which are tight enough to ensure continuity of the SLEκ trace.


Supervisor:  Not available  Sponsor:  EPSRC  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.707786  DOI:  
Keywords:  Gaussian free field ; Liouville quantum gravity ; Schramm Loewner evolution ; RohdeSchramm theorem ; probability ; geometry  
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