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SUMMARY:One-Dimensional Kardar-Parisi-Zhang (KPZ) Equation and its Univer
sality
DTSTART:20240301T130000Z
DTEND:20240301T150000Z
DTSTAMP:20240529T155800Z
UID:indico-event-11571@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Tomohiro Sasamoto (Tokyo Institute of Technology)\n\
nProbability and analysis informal seminarIn 1986\, Kardar\, Parisi and Zh
ang introduced a model equation for a growing surface\, in the form of a n
onlinear partial differential equation with noise[1]. In the original pape
r they applied a dynamical renormalization group analysis to demonstrate i
ts universal nature\, which is one of the first identified non-equilibrium
universality classes (KPZ universality class). Since then their equation
(KPZ equation) has been accepted as a standard model in non-equilibrium st
atistical mechanics. In this talk\, we focus on its one dimensional vers
ion because it has attracted particular attention in the last decade or so
. Mathematically there had been an issue of well-definendness of the equat
ion itself\, which was solved by a few different ideas. There is also a hi
gh precision experiment using liquid crystal. An important step was the di
scovery of an exact solution in 2010[2]\, which confirmed that the height
fluctuation is of O(t^(1/3)) and its universal distribution is given by th
e Tracy-Widom distribution from random matrix theory. Since then there hav
e been a large amount of studies on its generalizations\, which now forms
a field of “integrable probability”. The activity still continues. U
niversal behaviors for general initial conditions can now be studied (“K
PZ fixed point”). Very recently we have found a direct connection betwee
n KPZ systems and free fermion at finite temperature[3]. A remarkable as
pect of one dimensional KPZ is its unexpectedly wide universality. For exa
mple\, KPZ universality is expected to appear in long time behaviors of ma
ny one-dimensional Hamiltonian dynamical systems such as anharmonic chains
[4]. This is surprising because time-evolution of such systems are determ
inistic and there are apparently no growing surface with noise. More recen
tly people have observed appearance of KPZ behaviors in dynamical properti
es of quantum spin chains[5]\, first in numerical simulations but more rec
ently in real experiments. These discoveries have been attracting consider
able attention but theoretical foundations are not yet satisfactory. Refe
rences[1] M. Kardar\, G. Parisi\, and Y. C. Zhang\, Dynamic scaling of gro
wing interfaces\,Phys. Rev. Lett.\, 56\, 889–892 (1986).[2] T. Sasamoto
and H. Spohn\, One-dimensional Kardar-Parisi-Zhang equation: an exactsolut
ion and its universality\, Phys. Rev. Lett.\, 104:230602 (2010)\;G. Amir\,
I. Corwin\, and J. Quastel\, Probability distribution of the free energy
of the continuumdirected random polymer in 1+1 dimensions\, Comm. Pure App
l. Math.\, 64\, 466– 537 (2011).[3] T. Imamura\, M. Mucciconi\, T. Sasam
oto\, Solvable models in the KPZ class: approach throughperiodic and free
boundary Schur measures\, arxiv2204.08420. [4] H. Spohn\, Nonlinear fluc
tuating hydrodynamics for anharmonic chains\, J. Stat. Phys. 154\,1191–1
227 (2014).[5] M. Ljubotina\, M. Znidaric\, T. Prosen\, Kardar-Parisi-Zhan
g physics in the quantum Heisenberg magnet\,Phys. Rev. Lett. 122\, 210602
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e.\n\nhttps://indico.math.cnrs.fr/event/11571/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/11571/
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