Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions

Makoto Katori, Shinji Koshida*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

The present paper is concerned with properties of multiple Schramm-Loewner evolutions (SLEs) labelled by a parameter kappa is an element of (0, 8]. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the solution of the multiple Loewner equation generates multiple curves. To overcome this difficulty, we employ the coupling of Gaussian free fields and multiple SLEs. Consequently, we prove the longstanding conjecture that the solution indeed generates multiple continuous curves. Furthermore, these multiple curves are (i) simple disjoint curves when kappa is an element of (0, 4], (ii) intersecting curves when kappa is an element of (4, 8), and (iii) space-filling curves when kappa = 8.

Original languageEnglish
Article number325002
Number of pages19
JournalJournal of Physics A: Mathematical and Theoretical
Volume54
Issue number32
DOIs
Publication statusPublished - 13 Aug 2021
MoE publication typeA1 Journal article-refereed

Keywords

  • multiple Schramm-Loewner evolution
  • Dyson's Brownian motion model
  • Gaussian free field
  • GAUSSIAN FREE FIELDS
  • ERASED RANDOM-WALKS
  • CONFORMAL-INVARIANCE
  • PARTITION-FUNCTIONS
  • LOEWNER EVOLUTION
  • DUALITY
  • FORMULA

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