The Dynamical Sine-Gordon Model
File(s)
Accepted version
Author(s)
Hairer, M
Shen, H
Type
Journal Article
Abstract
We introduce the dynamical sine-Gordon equation in two space dimensions with parameter ββ, which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when β2∈(0,16π3)β2∈(0,16π3) the Wick renormalised equation is well-posed. In the regime β2∈(0,4π)β2∈(0,4π), the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for β2∈[4π,16π3)β2∈[4π,16π3), the solution theory is provided via the theory of regularity structures [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of 2+12+1 -dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al. [Commun Math Phys 337(2):569–632, 2015].
Date Issued
2015-12-21
Date Acceptance
2015-07-26
Citation
Communications in Mathematical Physics, 2015, 341 (3), pp.933-989
ISSN
0010-3616
Publisher
Springer Verlag
Start Page
933
End Page
989
Journal / Book Title
Communications in Mathematical Physics
Volume
341
Issue
3
Copyright Statement
The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-015-2525-3
Subjects
Science & Technology
Physical Sciences
Physics, Mathematical
Physics
KOSTERLITZ-THOULESS TRANSITION
ROUGHENING TRANSITION
WHITE-NOISE
COULOMB GAS
EQUATION
COLLAPSE
SYSTEMS
REGIONS
DIMENSIONS
Publication Status
Published