Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Remarks on the regularity of quasislits
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]

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: urn:nbn:se:kth:diva-263736OAI: oai:DiVA.org:kth-263736DiVA, id: diva2:1369299
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

Open Access in DiVA

fulltext(247 kB)4 downloads
File information
File name FULLTEXT01.pdfFile size 247 kBChecksum SHA-512
7bf20b25498eafd46f0be16863d2d9a81bfa5c18288be403b39fbd97cd8a4394a161cf2c09ee122b1e80fc737256f94163cbd7a7ef4db76aa02e787eb8576acb
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Schoug, LukasViklund, Fredrik
By organisation
Mathematics (Div.)
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 4 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 14 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf