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Dimension of two-valued sets via imaginary chaos
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-2547-6059
Université Lyon 1.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(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: urn:nbn:se:kth:diva-263735OAI: oai:DiVA.org:kth-263735DiVA, id: diva2:1369294
Note

QC 20191112

Available from: 2019-11-11 Created: 2019-11-11 Last updated: 2019-11-13Bibliographically approved
In thesis
1. On two-dimensional conformal geometry related to the Schramm-Loewner evolution
Open this publication in new window or tab >>On two-dimensional conformal geometry related to the Schramm-Loewner evolution
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
Conformal invariance, scaling, random fractals
National Category
Mathematics Mathematical Analysis Probability Theory and Statistics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-263774 (URN)978-91-7873-387-3 (ISBN)
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

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