Reflected Brownian Motions in the KPZ Universality Class

121 Pages·2017·1.755 MB·English

by  Thomas Weiss, Patrik Ferrari, Herbert Spohn (auth.)

SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 18
Thomas Weiss
Patrik Ferrari
Herbert Spohn
Reflected
Brownian
Motions in the
KPZ Universality
Class
123
SpringerBriefs in Mathematical Physics
Volume 18
Series editors
Nathanaël Berestycki, Cambridge, UK
Mihalis Dafermos, Cambridge, UK
Tohru Eguchi, Tokyo, Japan
Atsuo Kuniba, Tokyo, Japan
Matilde Marcolli, Pasadena, USA
Bruno Nachtergaele, Davis, USA
SpringerBriefs are characterized in general by their size (50-125 pages) and fast
production time (2-3 months compared to 6 months for a monograph).
Briefsareavailableinprintbutareintendedasaprimarilyelectronicpublicationto
be included in Springer’s e-book package.
Typical works might include:
(cid:129) An extended survey of a field
(cid:129) A link between new research papers published in journal articles
(cid:129) Apresentation ofcoreconceptsthatdoctoral students mustunderstand inorder
(cid:129) to make independent contributions
(cid:129) Lecture notes making a specialist topic accessible for non-specialist readers.
SpringerBriefsinMathematicalPhysicsshowcase,inacompactformat,topicsof
current relevance in the field of mathematical physics. Published titles will
encompassallareasoftheoreticalandmathematicalphysics.Thisseriesisintended
for mathematicians, physicists, and other scientists, as well as doctoral students in
related areas.
More information about this series at http://www.springer.com/series/11953
Thomas Weiss Patrik Ferrari
(cid:129)
Herbert Spohn
fl
Re ected Brownian Motions
in the KPZ Universality Class
123
ThomasWeiss Herbert Spohn
ZentrumMathematik ZentrumMathematik, M5
Technische UniversitätMünchen Technische UniversitätMünchen
Garching Munich
Germany Germany
Patrik Ferrari
Institut für Angewandte Mathematik
UniversitätBonn
Bonn
Germany
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
ISBN978-3-319-49498-2 ISBN978-3-319-49499-9 (eBook)
DOI 10.1007/978-3-319-49499-9
LibraryofCongressControlNumber:2016958720
©TheAuthor(s)2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
foranyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Inournotes,westudyamodelofinteractingone-dimensionalBrownianmotionsin
considerabledetail.Atfirstsight,thedynamicslooksimple:theBrownianmotions
are ordered, and Brownian motion with label j is reflected from the one with label
j(cid:1)1. This model belongs to the Kardar–Parisi–Zhang universality class. Adding
to the results as presented in our notes a recent contribution by Kurt Johansson,
this model allows for the so far most complete asymptotic analysis. The limiting
expressions are likely to be valid for all models in the KPZ class.
The notes are based on two joint publications and the Ph.D. Thesis of Thomas
Weiss. Our research was partially supported by the DFG under the grant numbers
SP181/29-1 and SFB1060, project B04.
Bonn, Germany Thomas Weiss
Munich, Germany Patrik Ferrari
September 2016 Herbert Spohn
v
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... . 1
2 One-Sided Reflected Brownian Motions and Related Models .. .... . 9
3 Determinantal Point Processes.. .... .... .... .... .... ..... .... . 25
4 Airy Processes.. .... .... ..... .... .... .... .... .... ..... .... . 31
5 Packed and Periodic Initial Conditions ... .... .... .... ..... .... . 45
6 Stationary Initial Conditions ... .... .... .... .... .... ..... .... . 71
7 More General Initial Conditions and Their Asymptotics. ..... .... . 97
vii
Chapter 1
Introduction
Back in 1931 Hans Bethe diagonalized the hamiltonian of the one-dimensional
Heisenbergspinchainthroughwhatisnowcalledthe“Betheansatz”(Bethe1931).
Atthattimephysicistswerebusywithotherimportantdevelopmentsandhardlyreal-
izedthemonumentalstep:forthefirsttimeastronglyinteractingmany-bodysystem
hadbeen“solvedexactly”.Inthe1960sLiebandLiniger(1963),Yang(1967),and
manymore (Sutherland2004)discoveredthatotherquantumsystemscanbehandled
via Bethe ansatz, which triggered a research area known as quantum integrability.
More details on the history of the Bethe ansatz can be found in Batchelor (2007).
EvenwiththeBetheansatzatone’sdisposal,itisahighlynon-trivialtasktoarriveat
predictionsofphysicalinterest.Thisiswhyeffortsinquantumintegrabilitycontinue
even today, reenforced by the experimental realization of such chains through an
arrayofcoldatoms(Simonetal.2011).
On a mathematical level, quantum hamiltonians and generators of Markov
processeshaveacomparablestructure.ThusonecouldimaginethattheBetheansatz
is equally useful for interacting stochastic systems with many particles. The first
indicationcamesomewhatindirectlyfromtheKardar-Parisi-Zhangequation(Kar-
daretal.1986),forshortKPZ,inonedimension.Werefertobooks(Barabasiand
Stanley1995;Meakin1998),lecturenotes(Johansson2006;Spohn2006;Quastel
2011; Borodin and Gorin 2012; Borodin and Petrov 2014; Spohn 2015), and sur-
veyarticles(Halpin-HealyandZhang1995;Krug1997;SasamotoandSpohn2011;
Ferrari and Spohn 2011; Corwin 2012; Takeuchi 2014; Quastel and Spohn 2015;
Halpin-HealyandTakeuchi2015).TheKPZequationisastochasticPDEgoverning
thetime-evolutionofaheightfunctionh(x,t)atspatialpointx andtimet,h ∈ R,
x ∈R,t ≥0.Theequationreads
∂ h = 1(∂ h)2+ 1∂2h+W (1.1)
t 2 x 2 x
withW(x,t)normalizedspace-timewhitenoise.Weusehereunitsofheight,space,
and time such that all coupling parameters take a definite value. For a solution
of (1.1), the function x (cid:4)→ h(x,t) is locally like a Brownian motion, which is
too singular for the nonlinearity to be well-defined as written. This difficulty was
©TheAuthor(s)2017 1
T.Weissetal.,ReflectedBrownianMotionsintheKPZUniversalityClass,
SpringerBriefsinMathematicalPhysics,DOI10.1007/978-3-319-49499-9_1
2 1 Introduction
resolvedthroughtheregularitystructuresof Hairer(2013),seealsoGubinelliand
Perkowski(2015)forasomewhatdifferentapproach.Kardar(1987)notedonelinkto
quantumintegrability.Heconsideredthemomentsofehandestablishedthattheyare
relatedtotheδ-Bosegaswithattractiveinteractions,whichisanintegrablequantum
many-bodysystem(LiebandLiniger1963).Moreprecisely,onedefines
(cid:2)(cid:3)n (cid:4)
E eh(yα,t) = f (y(cid:6)) (1.2)
t
α=1
with y(cid:6)=(y ,...,y ).Then
1 n
∂ f =−H f , (1.3)
t t n t
where H isthen-particleLieb-Linigerhamiltonian
n
(cid:5)n (cid:5)n
Hn =−21 ∂y2α − 21 δ(yα−yα(cid:8)). (1.4)
α=1 α(cid:7)=α(cid:8)=1
Almostthirtyyearslaterthegeneratoroftheasymmetricsimpleexclusionprocess
(ASEP) was diagonalized through the Bethe ansatz. In case of N sites, the ASEP
configuration space is {0,1}N signalling a similarity with quantum spin chains. In
fact, the ASEP generator can be viewed as the Heisenberg chain with an imagi-
naryXY-coupling.Forthetotallyasymmetriclimit(TASEP)andhalffilledlattice,
GwaandSpohn(1992)establishedthatthespectralgapofthegeneratorisoforder
N−3/2.ThesameorderisarguedfortheKPZequation.Thisledtothestrongbelief
that,despitetheirverydifferentset-up,bothmodelshavethesamestatisticalprop-
ertieson large space-time scales.Intheusual jargon of statisticalmechanics, both
modelsareexpectedtobelongtothesameuniversalityclass,baptizedKPZuniver-
salityclassaccordingtoitsmostprominentrepresentative.
The KPZ equation is solved with particular initial conditions. Of interest are
(i) sharp wedge, h(x,0) = −c |x| in the limit c → ∞, (ii) flat, h(x,0) = 0,
0 0
and (iii) stationary, h(x,0) = B(x) with B(x) two sided Brownian motion. The
quantityofprimeinterestisthedistributionofh(0,t)forlarget.Moreambitiously,
but still feasible in some models, is the large time limit of the joint distribution of
{h(xα,t),α=1,...,n}.
Inournotesweconsideranintegrablesystemofinteractingdiffusions,whichis
governedbythecoupledstochasticdifferentialequations
dxj(t)=βe−β(xj(t)−xj−1(t))dt +dBj(t), (1.5)
where{B (t)}acollectionofindependentstandardBrownianmotions.Forthepara-
j
meterβ >0wewilleventuallyconsideronlythelimitβ →∞.Butforthepurpose
of our discussion we keep β finite for a while. In fact there is no choice, no other
systemofthisstructureisknowntobeintegrable.Theindexsetdependsontheprob-
lem,mostlywechoose j ∈Z.Notethatx interactsonlywithitsleftindexneighbor
j
1 Introduction 3
xj−1. The drift depends on the slope, as it should be for a proper height function.
Buttheexponentialdependenceonxj−xj−1isveryspecial,howeverfamiliarfrom
otherintegrablesystems.ThefamousTodachain(Toda1967)isaclassicalintegrable
systemwithexponentialnearestneighbourinteraction.Itsquantizedversionisalso
integrable(Sutherland1978).
Interactionwithonlytheleftneighborcorrespondstothetotalasymmetricversion.
Partialasymmetrywouldread
(cid:6) (cid:7)
dxj(t)= pβe−β(xj−xj−1)−(1− p)βe−β(xj+1−xj) dt +dBj(t) (1.6)
with0≤ p ≤1.Thesearenon-reversiblediffusionprocesses.Onlyinthesymmetric
case, p = 1, the drift is the gradient of a potential and the diffusion process is
2
reversible.ThenthemodelisnolongerintheKPZuniversalityclassandhasvery
distinct large scale properties, see Guo et al. (1988); Chang and Yau (1992), for
example.
Equations(1.5)and(1.6)shouldbeviewedasadiscretizationof (1.1).Theinde-
pendent Brownian motions are the natural spatial discretization of the white noise
W(x,t). For the drift one might have expected the form (xj − xj−1)2 + xj+1 −
2xj +xj−1,butintegrabilityforcesanotherdependenceontheslope.For p =1the
similaritywiththeKPZequationisevenstrongerwhenconsideringtheexponential
moments
(cid:2)(cid:3)n (cid:4)
E eβxmα(t) = ft(m(cid:6)), (1.7)
α=1
m(cid:6) ∈Zn.Differentiatingint oneobtains
(cid:5)n (cid:5)n
d
β−2dt ft(m(cid:6))= ∂αft(m(cid:6))+ 21 δ(mα−mα(cid:8))ft(m(cid:6))−nft(m(cid:6)), (1.8)
α=1 α,α(cid:8)=1
whereδ istheKroneckerdeltaand∂αf(m(cid:6)) = f(...,mα,...)− f(...,mα −1,...).
Whencomparingwith(1.4),insteadof∂αonecouldhaveguessedthediscreteLapla-
cian(cid:2)α = −∂αT∂α withT denotingthetranspose.Toobtainsucharesult,thedrift
in (1.5) would have to be replaced by βe−β(xj−xj−1) +βe−β(xj−xj+1). But the linear
equationsfortheexponentialmomentsarenolongerBetheintegrable.Notehowever
thatthesemigroupsexp[∂αt]andexp[(cid:2)αt]differonalargescaleonlybyauniform
translation proportional to t. With such a close correspondence one would expect
system(1.5)tobeintheKPZuniversalityclassforanyβ >0,ashasbeenverifiedto
someextent(Amiretal.2011;SasamotoandSpohn2010;Borodinetal.2015).Also
partialasymmetry, 1 < p < 1,shouldbeintheKPZuniversalityclass.Infact,the
2
trueclaimisbymanyordersmoresweeping:theexponentialin(1.6), p (cid:7)= 1,canbe
2
replacedby“any”functionofxj−xj−1,exceptforthelinearone,andthesystemis
stillintheKPZuniversalityclass.Theredoesnotseemtobeapromisingideaaround
ofhowtoprovesuchaproperty.Currenttechniquesheavilyrelyonintegrability.