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On two-dimensional conformal geometry related to the Schramm-Loewner evolution
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-2547-6059
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three papers, one introductory chapter and one chapter with overviews of the papers and some additional results. The topic of this thesis is the geometry of models related to the Schramm-Loewner evolution.

In Paper I, we derive a multifractal boundary spectrum for SLEκ(ρ) processes with κ<4 and ρ chosen so that the curves hit the boundary. That is, we study the sets of points where the curves hit the boundary with a prescribed ``angle'', and compute the Hausdorff dimension of those sets. We study the moments of the spatial derivatives of the conformal maps gt, use Girsanov's theorem to change to an appropriate measure, and use the imaginary geometry coupling to derive a correlation estimate.

In Paper II, we study the two-valued sets of the Gaussian free field, that is, the local sets such the associated harmonic function only takes two values. It turns out that the real part of the imaginary chaos is large close to these sets. We use this to derive a correlation estimate which lets us compute the Hausdorff dimensions of the two-valued sets.

Paper III is dedicated to studying quasislits, that is, images of the segment [0,i] under quasiconformal maps of the upper half-plane into itself, fixing ∞, generated by driving the Loewner equation with a Lip-1/2 function. We improve estimates on the cones containing the curves, and hence on the Hölder regularity of the curves, in terms of the Lip-1/2 seminorm of the driving function.

Abstract [sv]

Denna avhandling består av tre artiklar, ett introduktionskapitel och ett kapitel där artiklarnas huvudresultat och bevisstrategi redovisas översiktligt.

I Artikel I härleder vi ett multifraktalt randspektrum för SLEκ(ρ)-processer med κ<4 och ρ vald så att kurvorna träffar randen. Vi studerar mängderna av punkter där kurvan träffar randen med en speciell ``vinkel'' och beräknar Hausdorffdimensionen av dessa mängder. Detta görs genom att studera spatiella derivator av de konforma avbildningarna gt, använda Girsanovs sats, samt använda ``imaginary geometry''-kopplingen för att hitta en korrelationsuppskattning.

I Artikel II studerar vi så kallade ``two-valued sets'' (TVS) för Gaussiska fria fält, det vill säga, lokala mängder sådana att den associerade harmoniska funktionen endast kan ta två värden. Det visar sig att realdelen av ``imaginary chaos'' är stor nära dessa mängder. Detta använder vi för att hitta en korrelationsuppskattning, vilken vi använder för att beräkna Hausdorffdimensionerna av TVS.

I Artikel III studerar vi kvasikonforma kurvor, genererade av Loewnerekvationen med drivfunktioner som är Lip-1/2. Vi förbättrar uppskattningarna för konerna i vilka de genererade kurvorna kommer att befinna sig och får genom detta bättre Hölderregularitet för dem, i termer av Lip-1/2-seminormen.

Place, publisher, year, edition, pages
Stockholm, Sweden.: KTH Royal Institute of Technology, 2019. , p. 67
Series
TRITA-SCI-FOU ; 2019;57
Keywords [en]
Conformal invariance, scaling, random fractals
National Category
Mathematics Mathematical Analysis Probability Theory and Statistics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-263774ISBN: 978-91-7873-387-3 (print)OAI: oai:DiVA.org:kth-263774DiVA, id: diva2:1369793
Public defence
2019-12-06, F3, Lindstedtsvägen 26, Stockholm, 09:00 (English)
Opponent
Supervisors
Available from: 2019-11-14 Created: 2019-11-13 Last updated: 2019-11-14Bibliographically approved
List of papers
1. A multifractal boundary spectrum for SLEκ(ρ)
Open this publication in new window or tab >>A multifractal boundary spectrum for SLEκ(ρ)
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study SLEκ(ρ) curves, with κ and ρ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed ``angle'' and determine the almost sure Hausdorff dimension of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps gt, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-263732 (URN)
Note

QC 20191112

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2019-11-13Bibliographically approved
2. Dimension of two-valued sets via imaginary chaos
Open this publication in new window or tab >>Dimension of two-valued sets via imaginary chaos
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Two-valued sets are local sets of the two-dimensional Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [-a,b]. Two-valued sets exist whenever a+b ≥ 2λ, where λ depends of the normalization of th.e GFF. We prove that the almost sure Hausdorff dimension of the two-valued set A-a,b equals d = 2-2λ2/(a+b)2. For the two-point estimate, we use the real part of a ``vertex field'' built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial d-dimensional measure supported on A-a,b and discuss its relation with the d-dimensional conformal Minkowski content for A-a,b

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-263735 (URN)
Note

QC 20191112

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2019-11-13Bibliographically approved
3. Remarks on the regularity of quasislits
Open this publication in new window or tab >>Remarks on the regularity of quasislits
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A quasislit is the image of a vertical line segment [0,iy], y>0, under a quasiconformal homeomorphism of the upper half-plane fixing ∞. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-1/2 class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-1/2 seminorm, σ. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the full range σ<4. The estimate is shown to be sharp for small σ. As qonsequences, we derive explicit Hölder exponents for σ<4 as well as estimates on winding rates. We also relate quantitatively the Lip-1/2 seminorm with the quasiconformal dilation and discuss the optimal regularity of quasislits achievable through reparametrization.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-263736 (URN)
Note

QC 20191112

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2019-11-13Bibliographically approved

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