Large deviations for the Kardar-Parisi-Zhang equationand the edge of random matrices
height fieldspace-time
white noiseLarge deviations for the Kardar-Parisi-Zhang equationand the edge of random matrices
height fieldspace-time
white noise

Cole-Hopf
SHELarge deviations for the Kardar-Parisi-Zhang equationand the edge of random matrices
height fieldspace-time
white noise

Cole-Hopf
SHE

P(h,t) ?

- typical fluctuations-tails of the PDF :large deviations (LD)exact results for LD1)
2)

thanks to exact solution for KPZ for all time t1601.05957, PLD, Majumdar,Schehr
1703.03310, Sasarov, Meerson, Prolhac

via Painleve

1802.08618, AK,PLD 1808.07710 AK,PLD, Prolhac via Cumulants
1803.05887, AK,PLD,Corwin, Ghosal, Tsai

via SAO, rigorous

1809.03410, Tsai

Alexandre
Krajenbrink

via CG

1811.00509, AK,PLD 4 methods (CG,SAO,Painleve,cumulants)

1802.03273, 1810.07129, Corwin, Ghosalrigorous

1603.03302, PLD, Majumdar,Schehr, Rosso
1705.04654, 1804.08800 AK,PLD

dropletBrownian, 1/2 space

1808.07710 AK,PLD, Prolhac expansion up to t^3 + large t guessOptimal fluct th. WNTKorshunov et al. (2007)
Meerson et al. 1512.04910+…

Numerics 1802.02106 Hartmann et al.
Hartmann, AK, PLD in preparation

KPZ equation:
known results for typical fluctuations for t >> 1

random variablewith Tracy-Widom (TW) distribution
PDF of largest eigenvalue of a random matrix
depend on class of initial condition h(x,0)

KPZ equation:
known results for typical fluctuations for t >> 1

Initial condition
1) Flat

GOE-TW

2) NWedge

GUE-TWh(x, t)

curved/droplet

t2 > t1

xt1 > 0t=03) Brownian

Baik-Rains

stationary

CDF of TW are given by- Fredholm determinants- Solutions of Painleve equations

KPZ equation:
known results for typical fluctuations for t >> 1

Initial condition
1) Flat

GOE-TW

2) NWedge

GUE-TWh(x, t)curved/droplet

t2 > t1x
t1 > 0t=03) Brownian
stationary

Question: tails of P(H,t) ?

Baik-Rains

Tails of P(H,t) ?

short time

Tails of P(H,t) ?

short time
Edwards-Wilkinson equation (EW)

1) typical fluctuations

KPZeq => EWeq
GaussianTails of P(H,t) ?

short time
Edwards-Wilkinson equation (EW)

1) typical fluctuations

KPZeq => EWeq
Gaussian2) large deviations

depends on IC

Tails of P(H,t) ?

short time
Edwards-Wilkinson equation (EW)

1) typical fluctuations

KPZeq => EWeq
Gaussian2) large deviations

2 methods:
exact solutions -> exact
weak noise theory/optimal fluctuation-> exact tails, diff. eq. numericaldepends on IClarge time

“typical” tails= the tails of TW distributions universal

large time

“typical” tails= the tails of TW distributions universal

right taildroplet,flat

brownianlarge time

“typical” tails= the tails of TW distributions universal

right taildroplet,flat

brownianleft taildroplet, brownian
cubic tailPainleve IIflat

large time

“typical” tails= the tails of TW distributions universal

right taildroplet,flat

brownianleft taildroplet, brownian
cubic tailPainleve IIflat
Q: what are far tails (large deviations) ?
does the cubic TW tail extend to all H,t ?

prediction

large deviation regime

1601.05957, PLD, Majumdar,Schehrprediction

large deviation regime

1601.05957, PLD, Majumdar,Schehr

non-universal
universal

prediction

large deviation regime

1601.05957, PLD, Majumdar,Schehr

non-universal
universal

analogy withlarge deviation for random matrices

prediction

large deviation regime

1601.05957, PLD, Majumdar,Schehr

non-universal
universaltlargest eigenvalue
complex whishart matrix(x=0,t)H <<typicalproba all sites in square
have small energy

Johansson (2000)
H >> typicalproba all sites along blue line
have large energyx(0,0)

Matching TW tails with large deviations

TW tailMatching TW tails with large deviations

TW tail=

Large deviations
universal

Matching TW tails with large deviations

TW tail=

Large deviations
universal=universal

Matching TW tails with large deviations

TW tail=

Large deviations
universal=

rate functions for KPZ equation?

universal

prediction

large deviation regime

1601.05957, PLD, Majumdar,Schehr

for the KPZ equation
droplet IC

prediction

large deviation regime

1601.05957, PLD, Majumdar,Schehr

for the KPZ equation
droplet IC

finite time solution of droplet IC
non-local extension of Painleve II equation
Amir, Corwin, Quastel (2010)

prediction

large deviation regime1601.05957, PLD, Majumdar,Schehr

for the KPZ equation
droplet IC

wrong !
finite time solution of droplet IC

1703.03310, Sasarov, Meerson, Prolhac

non-local extension of Painleve II equation
Amir, Corwin, Quastel (2010)

semi-classical ansatz

prediction

large deviation regime

1601.05957, PLD, Majumdar,Schehr

for the KPZ equation
droplet IC

wrong !
finite time solution of droplet IC

1703.03310, Sasarov, Meerson, Prolhac

non-local extension of Painleve II equation

semi-classical ansatz

Amir, Corwin, Quastel (2010)

now: 4 methods now to solve the problem

calculation of

from exact solution for KPZ

droplet ICcalculation of

from exact solution for KPZ

droplet IC=
Calabrese,PLD,Rosso;Dotsenko
Spohn,Sasamoto;Amir,Corwin,Quastel (2010)

calculation of

from exact solution for KPZ

droplet IC=
Borodin,Gorin (2016)

calculation of

from exact solution for KPZ

droplet IC=
Borodin,Gorin (2016)

Airy point process
determinantal point process
n-point correlations=mean density

GUE [-2,2] eigenvalues

0
APP is scaled edge limit of GUE

calculation of

from exact solution for KPZ=(0)
Airy point process

fixed(0)=typical fluctuations(1)
mean density

droplet ICcalculation of

from exact solution for KPZ

droplet IC=(0)
Airy point process

fixed(0)=(1)
mean density=(1) =
GUE-TWtypical fluctuations

calculation of

from exact solution for KPZ

droplet IC=(0)
Airy point process

fixed(0)=(1)mean density=(1) =
GUE-TWtypical fluctuations
large deviations:

=(0)

(1)
=

mean density

=(0)

(1)
== (1) =mean density

=(0)

(1)
== (1) =mean density

=(0)

(1)
== (1) ==mean density

=(0)

(1)
== (1) ==>=mean density

=(0)

(1)
== (1) ==>rules out |z|^3=

mean density

=(0)

(1)
== (1) ==>rules out |z|^3=

far tail same for small and large time !

mean density

large timeleft tail H/t<0?

1st cumulant

2nd cumlarge timeleft tail H/t<0?

1st cumulant

2nd cum

1) Non-local Painleve eq

1703.03310, Sasarov, Meerson, Prolhac

2) Coulomb gas (edge)

1803.05887, AK,PLD,Corwin, Ghosal, Tsai

3) Stochastic Airy Operator

1809.03410, Tsai rigorous

4) Cumulant expansion

1802.08618, AK,PLD
1808.07710 AK,PLD, Prolhac

Also 1811.00509, AK,PLD 4 methods (CG,SAO,Painleve,cumulants)
1802.03273, 1810.07129, Corwin, Ghosalrigorousvia the Coulomb gas at the edge
rescaled APP

empirical measure+..

via the Coulomb gas at the edge
rescaled APP

empirical measure

LDP for AiryPP+..

via the Coulomb gas at the edge
rescaled APP

LDP for AiryPP

empirical measure=>

variational problem+..

via the Coulomb gas at the edge
empirical measure

rescaled APP

LDP for AiryPP=>

variational problem

obtained from LDP for GUE
taking edge limit

+..

bz0

bz

0=

bz
hard wall0
Dean,Majumdar=

short time large deviations

droplet, flat, Brownien

1/2space x>0

short time large deviations

droplet, flat, Brownien

1/2space x>0

droplet, Brownien
flat
1/2 space drop

perturb theodroplet, flat
Brownien

exact solutions -> exact
weak noise theory/optimal fluctuation-> exact tails, diff. eq. numericalsaddle point

strictly convex

higher cumulant
first cumulant

dropletBrownian

Lambert functions

higher cumulant
first cumulant

dropletBrownian

Lambert functions

Exact short-time height distribution in 1D KPZ equationand edge fermions at high temperature

PLD, S. Majumdar, A. Rosso, G. Schehr,
Phys. Rev. Lett. 117 070403 (2016).

A. Krajenbrink, PLD, Phys. Rev. E 96, 020102 (2017)

z is in
H is inz is in
H is inz is in
H is inNumerical Simulations

directed polymer
droplet IC/full space
no free parametersquare lattice T >>1=> KPZ equation

importance samplingleft LD tailright LD tailBrownian and flat IC/full space

Hartmann, AK, PLD in preparation

later times
droplet IC/full spacefor the KPZ equation
1) large time large deviation (left)
droplet IC obtained

how to obtain general IC ?

A=-1/2, GOE

can one bypass exact solutions ?

2) short time large deviation

obtained
systematic expansion up to O(t^3)

connection to Weak Noise Theory ? integrability?

multi-point?

far tails large |H| seem same large t and small t ? process?