Large deviations for the height in 1D Kardar-Parisi-Zhang growth at late times

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by  Pierre Le Doussal

Large deviations for the height in 1D Kardar-Parisi-Zhang growth at late times
Pierre Le Doussal,1 Satya N. Majumdar,2 and Gr´egory Schehr2
1CNRS-Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris Cedex, France
2LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France
(Dated: January 25, 2016)
We study the atypically large deviations of the height H ∼ O(t) at the origin at late times in
1+1-dimensional growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class.
We present exact results for the rate functions for the discrete single step growth model, as well
as for the continuum KPZ equation in a droplet geometry. Based on our exact calculation of the
6 rate functions we argue that models in the KPZ class undergo a third order phase transition from
1
a strong coupling to a weak coupling phase, at late times.
0
2

n The celebrated Tracy-Widom (TW) distribution was P(H,t)
a discovered originally in random matrix theory (RMT) Tracy-Widom
J
[1, 2]. In RMT, it describes the probability distribution

2 typical fluctuations
of the typical fluctuations of the largest eigenvalue of a
2
Gaussian random matrix. Since then, this distribution
t1/3
] has emerged in a variety of problems [3, 4] (unrelated a
h
priori to RMT), ranging from random permutations [5]
c
all the way up to the Yang-Mills gauge field theory [6].
e
m Why is TW distribution so ubiquitous? It was recently
- shown that in several systems where TW distribution ⇠ e�t2��(H/t) ⇠ e�t�+(H/t)
t occurs there is usually an underlying third order phase
a right large
t transition between a strong and a weak coupling phase left large deviation tail
s
[7]. In these systems the TW distribution appears as a deviation tail
.
t
a finite-size crossover function connecting the free-energies
m ofthetwophasesacrossthethirdordercriticalpoint[7].
H
- In the strong coupling phase, the degrees of freedom of ⇠O(t) ⇠O(t)
d
thesystemactcollectivelywhiletheweakcouplingphase
n
is described by a single dominant degree of freedom. In FIG. 1. A schematic picture of the height distribution at the
o origin. ThetypicalfluctuationsH ∼O(t1/3)aroundthemean
c the context of RMT, this third-order phase transition
aredistributedaccordingtotheTracy-WidomGUElaw(blue
[ shows up in the distribution of the top eigenvalue λ
max line). Theatypicallargefluctuationstotheleft(redline)and
of a N N matrix belonging to the classical Gaussian
1 × totheright(greenline)aredescribedrespectivelybytheleft
v ensembles[7]. Thecentralpartofthedistribution,corre- and right large deviation functions in Eq. (1).
7 sponding to the typical fluctuations of λ , is described
max
5 by the TW distribution, while the atypically large fluc-
9
tuations to the left (right) correspond to the strong (re-
5
spectively weak) coupling phases. at late times t 1. Specifically, we find that P(H,t),
0 (cid:29)
for t 1, has three different behaviors
1. For a wide class of 1+1-dimensional interface growth (cid:29)
0  e t2Φ (H/t) , H (t)<0 I
:16 mvfleuorcsdtaeullaisttybioecnlloasnsgsgri[on8wg],tiaotttishlaewteKellatirkmdnaeorsw-Pansatrhisait-tZ1t/hh3aen[t9gy].p(iKcMaPloZhr)eeouigvnheirt-  −1 −(cid:20) H (cid:21) ∼O
v ∼ P(H,t) f , H (t1/3) II (1)
arXi tcThahleiflspurTcotWbuaabdtiilioistntyrsidbaiursettirgoiibnvuehtniaobsnyaftlushnoecbTtieoWenn(dvPiesDtrriFfiib)eudotfieotxhnpe[es1re0imt–y1ep9ni]--. ∼ et−1/t3Φ+(Ht1//t)3 , H ∼O(t)>0 III.
tallyinliquidcrystalandpaperburningsystems[20–22]. ∼O
The appearance of the TW distribution in these growth The regime II is well known and it describes the typical
models then raises a natural question: is there a third height fluctuations (H (t1/3)) and the scaling func-
∼ O
order phase transition between a strong and a weak cou- tion f(s) is given by the TW distribution. The scaling
pling phase in such growth models? If so, how can one function depends on the initial conditions: for the flat
describe these two phases? In this Letter, we show that geometry it corresponds to f (s) (i.e., TW for theGaus-
1
there is indeed a third order phase transition in these sian Orthogonal Ensemble, GOE), while for the curved
growth models by studying the probability distribution (or droplet) geometry, it corresponds to f (s) (i.e., TW
2
P(H,t) of the height H at the origin (suitably centered) for the Gaussian Unitary Ensemble, GUE). These distri-
2
butions have asymmetric non-Gaussian tails: h
(cid:40)
fβ(s)∼ ee−−22β3β4|ss3|/32,,ss→−+∞ , (2) h(x,t2)
→ ∞
where β = 1 and 2 correspond respectively to GOE and
GUE.
The new results in this Letter concern the atypical
large height fluctuations in regime I and III in Eq. (1). h(x,t )
1
The regime I corresponds to the large negative fluctua-
tions (H (t) < 0) and is characterized by the left
∼ O
large deviation function Φ (z). Similarly, the regime III
−
describes the large positive fluctuations (H (t) > 0)
∼ O
andischaracterizedbytherightlargedeviationfunction
Φ+(z). These two rate functions Φ±(z) are the char- �t2 �t1 x=0 t1 t2 x
acteristics of the two phases: Φ (z) corresponds to the
−
strong coupling phase, while Φ (z) describes the weak
+ FIG. 2. The height h(x,t) evolving on a substrate −t ≤
coupling phase (as explained later). Note that on the
x ≤ t. The light cone (black lines) describes the evolution
scale H (t), the central part of width (t1/3) is ef-
of the substrate. The solid line (in blue) represents the av-
∼ O O
fectively reduced to a point z = 0 as t . Indeed, it erage height at two different times, (cid:104)h(x,t)(cid:105) = v(x/t)t with
→ ∞ √
follows from Eq. (1) that v(z)=1+ 1−z2 having a semi-circular shape.
(cid:40)
1 Φ (z), z 0
lim lnP(H =zt,t)= − ≤ (3)
t −t2 0, z 0.
→∞ ≥ exhibiting a semi-circular droplet shape (see Fig. 2).
Thusast , z =0becomesacriticalpointandΦ (z) Moreover the height at the origin at late times behaves
canbeinte→rp∞retedasthe“freeenergy”ofthestrong−cou- as h(0,t) 2t+2t1/3χ2, where χ2 is a t-independent
(cid:39)
pling phase. We further show that it vanishes univer- random variable distributed via the TW distribution for
sally, Φ (z) z 3, as z 0−, thus indicating a third the GUE, f2(s) [10]. By exploiting an exact mapping
order ph−ase t∝ran|s|ition. Th→erefore in order to probe this to the largest eigenvalue of complex Wishart matrices
thirdordertransitionitisimportanttocomputethelarge [10], and using the results for the large deviations of the
deviation functions. In this Letter, we compute Φ (z) latter [24, 25], we establish the result in Eq. (1) (with
explicitly for the droplet geometry in (i) a discrete±sin- H = h(0,t) t). In regime I, we get [26]:
2 −
gle step growth model belonging to the KPZ class and
(ii) the continuum KPZ equation. In general, Φ (z) are Φ (z)= 1(cid:0)2z z2 2 ln(1+z)(cid:1) , 1<z 0,(5)
non-universal and depend on the model. Howev±er, their − 8 − − − ≤
small arguments behaviors are universal: Φ (z) z 3 where z > 1 since the height h(0,t) > 0. As z 0 ,
as z 0− and Φ+(z) z3/2 as z 0+.−Inde∝ed,| a|s one gets Φ−(z) z 3/12 as announced in the intr→odu−c-
the cr→itical point z = 0∝is approached→from either side, tion. In reg−ime I∼II|, w|e find
the large deviation behaviors smoothly match with the
asymptotic tails of the TW distribution (2). (cid:112) (cid:16) (cid:112) (cid:17)
Φ (z)=2 z(z+1)+ln 2z+1 2 z(z+1) , z 0,
+
We start by analyzing a directed polymer model be- − ≥
(6)
longing to the KPZ universality class studied by Johans-
which behaves as Φ (z) (4/3)z3/2 as z 0+. Note
son [10]. This model can be translated to a discrete + ∼ →
that in regime II, if we make H (t) and use the
space-time (x,t) growth model in a “droplet” geometry. ∼ O
asymptoticbehaviorsofTWdistributioninEq. (2)with
The growth takes place on the substrate t x t (see
− ≤ ≤ β = 2, it can be checked that it matches smoothly with
Fig. 2), starting from the seed at the origin x = 0 at
the large deviation regimes on both sides. Interestingly,
t=0. The interface height h(x,t), at site x and at time
in this height model (4) there is a clear physical expla-
t, evolves in the bulk t<x<t as [23]
− nation as to why the left tail [regime I in (1)] scales like
h(x,t)=max[h(x 1,t 1),h(x+1,t 1)]+η(x,t) (4) e−t2 while the right tail [regime III in (1)] behaves
− − − ∼like e t. Indeed, in order to realize a configuration
−
∼
where η(x,t) 0’s are independent and identically of H much smaller than its typical value (regime I), the
≥
distributed (i.i.d.) nonnegative random variables each noise variables η(x,t) at all the sites within the 1+1-
drawn from an exponential distribution: p(η) = e η for dimensionalwedge(cfFig. 2)shouldbesmall. Indeed,if
−
η 0. Johansson showed that at late times, the aver- any of the η(x,t) within this wedge is big, the dynamics
≥
age height h(x,t) = v(x/t)t with v(z) = 1+√1 z2 in Eq. (4) would force the neighboring sites at the next
(cid:104) (cid:105) −
3
time step to be big. The probability of this event, where To derive the rate functions for the continuum KPZ
collectivelyallthenoisevariablesη(x,t)insidethewedge case, we start from an exact formula [15–18], valid at all
(x < t), of area t2, are all small is proportional to times t in the droplet geometry. It relates the following
| | ∝
e t2 (the noise variables being i.i.d.). In contrast, a con- generating function to a Fredholm determinant (FD)
−
figuration where H is much bigger than its typical value
(regimeIII)canberealizedbyaddinglargepositivenoise gt(s):= exp( eH(t)−t1/3s) =det(I PsKtPs) (10)
(cid:104) − (cid:105) −
variables at the origin η(x=0,τ) at all times τ between
0 and t. The probability of this event is simply e t as where the finite time kernel is
−
∝
the noises at different times are i.i.d. Hence this event (cid:90)
∞ Ai(r+u)Ai(r(cid:48)+u)
is not a collective one, unlike the left large deviation. Kt(r,r(cid:48))= du 1+e t1/3u (11)
Thus, the left large deviation [regime I in Eq. (1)] is −∞ −
theanalogueofthe‘strongcouplingphase’andtheright and P is the projector on the interval [s,+ ) [30]. In
s
∞
large deviation [regime II in Eq. (1)] corresponds to the Eq. (11), Ai(x) denotes the Airy function.
‘weak coupling’ phase. The transition between the two Let us recall that to obtain the typical fluctuations
phases is a third order phase transition, as Φ (z) z 3 regime (II) in formula (1), where H(t) t1/3, one needs
as z 0−, as mentioned above. This pictu−re is∝v|er|y to take the limit t + at fixed s∼in (10). In that
simila→r to other third order phase transitions observed limit K (r,r ) conve→rges∞to the standard Airy kernel,
t (cid:48)
before in RMT and reviewed recently in Ref. [7]. KAi(r,r(cid:48)) = (cid:82)0∞duAi(r + u)Ai(r(cid:48) + u) and the right
WhiletherighttailratefunctionΦ+(z)hasbeenstud- hand side (r.h.s.) converges to the GUE-TW distribu-
ied numerically [27] and, more recently, analytically [28] tion. The left hand side (l.h.s.) of (10) converges to
in discrete growth models, the left tail Φ (z) is much θ(s t 1/3H(y)) (whereθ(x)istheHeavisidestepfunc-
−
− (cid:104) − (cid:105)
harder to compute, and there are very few exact results, tion), and one obtains
anexceptionbeingthelongestincreasingsubsequencein
randompermutations(forbothtails)[29]. Wenowshow lim Prob.(χ <s)=det(I P K P )=F (s) (12)
t s Ai s 2
t + −
that these rate functions can also be calculated for the → ∞
continuum KPZ equation itself, where the height field (cid:82)s
where F (s) = f (s)ds is the cumulative distribu-
2 2 (cid:48) (cid:48)
h(x,t) evolves as [8] tion function (C−D∞F) of the GUE-TW distribution. To
compute the rate functions Φ (z) we now consider the
∂ h=ν∂2h+ λ0 (∂ h)2+√Dξ(x,t), (7) formula (10) in the limit whe±n s and t are both large,
t x 2 x
keeping the ratio y =s/t2/3 fixed.
whereν >0isthecoefficientofdiffusiverelaxation,λ > Righttail. Westartwiththerightlargedeviationfunc-
0
0isthestrengthofthenon-linearityandξ(x,t)isaGaus- tion, therefore we consider formula (10) in the regime of
sian white noise with zero mean and ξ(x,t)ξ(x,t) = large s > 0. Consider first the l.h.s. of Eq. (10). It is
(cid:48) (cid:48)
δ(x x)δ(t t). Weuseeverywhereth(cid:104)enaturalunit(cid:105)sof convenient to introduce a random variable γ (indepen-
(cid:48) (cid:48)
spac−e x =−(2ν)3/(Dλ2), time t = 2(2ν)5/(D2λ4) and dent of H) distributed via the Gumbel distribution, of
∗ 0 ∗ 0
height h = 2ν. CDF given by
∗ λ0
Here for definiteness we focus on the narrow wedge
initial condition, h(x,0) = x/δ, with δ 1, which (cid:104)θ(b−γ)(cid:105)γ =e−e−b . (13)
−| | (cid:28)
gives rise to a curved (or droplet) mean profile as time Substituting b = st1/3 H in (13) allows us to rewrite
evolves [15–19]. We focus on the shifted height at the −
the l.h.s of (10) as
origin at x = 0, H(t) = h(0,t)+ t , which fluctuates
12
typicallyonascalet1/3 arounditsmeanatlargetime,as 1 exp( eH(t) t1/3s) = Prob(H >st1/3 γ) . (14)
− γ
described by the regime II in Eq. (1) with f(s) = f (s), −(cid:104) − (cid:105) (cid:104) − (cid:105)
2
the TW distribution for the GUE. Now consider the r.h.s of Eq. (10) for s 1. Expanding
(cid:29)
We show below that for the continuum KPZ equation, the FD in powers of K and keeping only the first two
t
inadropletgeometry, thegenericresultinEq. (1)holds terms one obtains
in regime I and III as well. Interestingly the rate func-
(cid:90) +
tions turn out to be rather simple in this case ∞
det(I P K P ) 1 drK (r,r). (15)
s t s t
− (cid:39) −
s
1
Φ (z)= z 3 , z 0 (8)
− 12| | ≤ Equating Eq. (14) and (15) and taking a derivative with
4 respect to s gives
Φ (z)= z3/2 , z 0. (9)
+
3 ≥
t1/3 P(H =st1/3 γ,t) =K (s,s) (16)
γ t
Thus the continuum KPZ equation also exhibits a third (cid:104) − (cid:105)
order phase transition at the critical point z =0. a relation exact for all t.
4
We first study the asymptotics of K (s,s) for large where
t
s t2/3. Performing a change of variable u = t2/3v,
∼ − 1
(11) becomes σt(v)= 1+e t1/3v (23)
−
K (yt2/3,yt2/3)=t2/3(cid:90) +∞dvAi2(t2/3(y−v)) (17) and σ (v) = ∂ σ (v). The function q (s,v) satisfies a
t 1+etv t(cid:48) v t t
non-linear integro-differential equation in the s variable
−∞
with y = (1). This integral can be analyzed for large t
O (cid:90) +
[26] and we obtain [31] ∂2q (s,v)=(s+v+2 ∞dwσ (w)[q (s,w)]2)q (s,v)
s t t(cid:48) t t
(cid:40)
K (yt2/3,yt2/3) e tI(y) , I(y)= 43y3/2, 0<y < 14 −∞ (24)
t ∼ − y 1 , y > 1 , with the boundary condition q (s,v) Ai(s+v).
− 12 4 t (cid:39)s→+∞
(18) Inthelonglimitt→+∞,σt(cid:48)(v)→δ(v)andhenceqt(s,0)
where the pre-exponential factors are given in [26]. Hav- satisfies the standard Painlev´e II equation [1].
ing obtained the r.h.s of (16) we now consider its l.h.s. For large but finite t, we substitute the anticipated
We anticipate (and verify a posteriori) that in this right scaling form Q (s) e t2Φ (y=s/t2/3) in (22). The con-
t ∼ − −
tail the PDF has the form (setting z =H/t) sistencythensuggeststhatq (s,v)takesthescalingform
t
4
lnP(H,t)=−t3z3/2−alnt−χdroplet(z)+o(1) (19) qt(s,v)(cid:39)t1/3q˜(s/t2/3,vt1/3), fort →∞, (25)
where the constant a and the function χ (z) are yet and the scaling function q˜(y,v) satisfies
droplet
to be determined. Inserting this form on the l.h.s. of
Eq. (16), analyzing the resulting integral [26] and com- (cid:90) +∞dvq˜(y,v)2e−v =Φ (y). (26)
(cid:48)(cid:48)
paring it to the r.h.s. in (18), we find that indeed the (1+e−v)2 −
−∞
ansatz in (19) is correct with a=1 and an explicit form
Substituting further the scaling form (25) in the differ-
for χ (z) given in Eq. (80) of the Supp. Mat. [26].
droplet ential equation (24) we obtain as t
Finally, keeping only the leading behavior of (19) gives →∞
us the exact right rate function (cid:90) +∞ q˜(y,v)2e−v
y+2 dv =0. (27)
4 (1+e v)2
Φ (z)= z3/2 , z 0, (20) −
+ −∞
3 ≥
Comparing with (26) immediately gives for all z 0,
as announced in Eq. (9). For the pre-exponential factor ≤
Φ (z) = z. Solving with the boundary condition
in the flat case we find a = 1/2 and χflat(z) given in Φ(cid:48)−(cid:48)(z) −2z 3/12,comingfrommatchingwiththeleft
Eq. (87) of the Supp. Mat. [26]. − (cid:39)z→0 | |
tail of the TW GUE distribution as z 0 , implies
This result is also consistent with the known exact → −
large time behavior of the moments, enH ∼t→+∞ e112n3t, Φ (z)= 1 z 3 , z 0, (28)
calculated using the Bethe ansatz [32]. Indeed a saddle − 12| | ≤
4( H )3/2
point calculation using P(H,t) e−3 t1/3 reads as announced in Eq. (8).
∼
(cid:90)
dH enH−43(t1H/3)3/2 ∼e112n3t (21) heIignhtsuamtmlaateryt,imouesr froersuglrtoswotnh mlaorgdeelsdeinviathtieonKsPfZorcltahses
suggest a third order phase transition between a strong
where the saddle point, at H = n2t/4 for fixed inte-
n and a weak coupling phase. Generically the associated
ger n, is precisely in the right large deviation regime.
ratefunctionsarenon-universalbuttheirsmallargument
Note that the dependence on the initial condition ap-
behavior are universal, as they match the TW tails. In
pears only in the (subdominant) pre-exponential factor
the case of the continuum KPZ equation these functions
of the moments, as discussed in [26] where we establish
aresimple,Eqs. (8,9),showingthattheTWuniversality
that Φ (z)= 4z3/2 both for droplet and flat initial con-
+ 3 extendsallthewaytothelargedeviationregime. Anat-
ditions.
uralquestionishowthislatetimebehaviorisapproached
Left tail. We now focus on the left tail where we set
as time increases. Weak noise expansion and instanton
H/t (1) < 0. In this case, one can show [26] that
∼ O calculationsinthetails(fortheflatgeometry)indicatea
the l.h.s. of (10) scales as e t2Φ (y=s/t2/3) for y =
O(1). The r.h.s of (10), Qt∼(s) :−= d−et(I −PsKtPs), is idnifftehreenetarblyehtaivmioerrPeg(iHm,et)t ∼ e1−[|H33|5,/32/4t]1./2Ininfatchtewleefthtaaviel
not easy to analyze in the regime of large negative s. (cid:28)
computed exactly the short time height distribution in
Fortunately in Ref. [18] the authors proved an exact
the droplet geometry which exhibits a similar H 5/2 left
differential equation satisfied by Qt(s): tail behavior [35], manifestly different from the| la|te time
∂s2lnQt(s)=−(cid:90) +∞dvσt(cid:48)(v)[qt(s,v)]2 (22) tbheeharvigiohrt t|Hail|3Ho3b/t2aiinseadlrehaedreyaatttlaaitneedtimatees.arlIyntciomnet.rast,
−∞
5
We thank D. Dean, B. Meerson, J. Quastel, H. Spohn Theor. 40, 4317 (2007).
and K. Takeuchi for useful discussions. We acknowl- [25] S. N. Majumdar, M. Vergassola, Phys. Rev. Lett. 102,
edge support from PSL grant ANR-10-IDEX-0001-02- 060601 (2009).
[26] P. Le Doussal, S. N. Majumdar, G. Schehr, see supple-
PSL (PLD). We thank the hospitality of KITP, under
mental material, which includes Refs. [36–39].
Grant No. NSF PHY11-25915.
[27] C. Monthus, T. Garel, Phys. Rev. E 74, 051109 (2006).
[28] N. Georgiou, T. Seppa¨la¨inen, Ann. Probab. 41, 4248
(2013); C. Janjigian, J. Stat. Phys. 160, 1054 (2015);
E. Emrah, C. Janjigian, arXiv:1509.02234.
[29] J. D. Deuschel, O. Zeitouni, Comb. Probab. Comput. 8,
[1] C.A.Tracy,H.Widom,Commun.Math.Phys.159,151 247 (1999).
(1994). [30] We recall that, for a trace-class operator K(x,y)
[2] C.A.Tracy,H.Widom,Commun.Math.Phys.177,727 such that TrK = (cid:82) dxK(x,x) is well defined,
(1996). det(I − K) = exp[−(cid:80)∞ TrKn/n], where TrKn =
[3] C.A.Tracy,H.Widom,Proceedings of the ICM Beijing, (cid:82) dx ···(cid:82) dx K(x ,x )Kn(=x1 ,x )···K(x ,x ). The ef-
1 n 1 2 2 3 n 1
1, 587 (2002). fect of the projector P in (10) is simply to restrict the
s
[4] For a recent popular article on the subject, see N. integrals over x ’s to the interval [s,+∞).
i
Wolchover, At the Far Ends of a New Universal Law, [31] In equation (18) the change of behavior at y = 1/4 can
Quanta magazine (October, 2014), available online be simply understood as follows. The expansion of the
at: https://www.quantamagazine.org/20141015-at-the- generating function in the left hand side of (10) gives
far-ends-of-a-new-universal-law . (cid:39) 1−e−t1/3s(cid:104)eH(cid:105) = 1−e−t1/3(y−1/12) where we used
[5] J. Baik, P. Deift, K. Johansson, J. Am. Math. Soc. 12,
that (cid:104)eH(cid:105) = et/12 and y = st−2/3. Taking a derivative
1119 (1999).
with respect to s then gives the second line of (18).
[6] P. J. Forrester, S. N. Majumdar, G. Schehr, Nucl. Phys.
[32] M. Kardar, Nucl. Phys. B 290, 582 (1987).
B 844, 500 (2011).
[33] B. Meerson, E. Katzav, A. Vilenkin, preprint
[7] S. N. Majumdar, G. Schehr, J. Stat. Mech. P01012
arXiv:1512.04910.
(2014).
[34] I. V. Kolokolov, S. E. Korshunov, Phys. Rev. E 80,
[8] M. Kardar, G. Parisi, Y.C. Zhang, Phys. Rev. Lett. 56,
031107 (2009); Phys. Rev. B 78, 024206 (2008); Phys.
889 (1986).
Rev. B 75, 140201 (2007).
[9] D. A. Huse, C. L. Henley, D.S. Fisher, Phys. Rev. Lett.
[35] To be published elsewhere.
55, 2924 (1985); T. Halpin-Healy, Y-C. Zhang, Phys.
[36] C.Nadal,S.N.MajumdarJ.Stat.Mech.P04001(2011).
Rep. 254, 215 (1995); J. Krug, Adv. Phys. 46, 139
[37] P.Calabrese,P.LeDoussal,Phys.Rev.Lett.106,250603
(1997).
(2011); P. Le Doussal, P. Calabrese, J. Stat. Mech.
[10] K. Johansson, Commun. Math. Phys. 209, 437 (2000).
P06001 (2012).
[11] M.Pra¨hofer,H.Spohn,Phys.Rev.Lett.84,4882(2000);
[38] J. Baik, R. Buckingham, J. DiFranco, Commun. Math.
J. Baik, E. M. Rains J. Stat. Phys. 100, 523 (2000).
Phys. 280, 463 (2008).
[12] J. Gravner, C. A. Tracy, H. Widom, J. Stat. Phys. 102,
[39] G. Borot, C. Nadal, Random Matrices: Theory Appl. 1,
1085 (2001).
1250006 (2012).
[13] S. N. Majumdar, S. Nechaev, Phys. Rev. E 69, 011103
(2004).
[14] S. N. Majumdar, S. Nechaev, Phys. Rev. E 72, 020901
(2005).
[15] T. Sasamoto, H. Spohn, Phys. Rev. Lett. 104, 230602
(2010).
[16] P. Calabrese, P. Le Doussal, A. Rosso, Europhys. Lett.
90, 20002 (2010).
[17] V. Dotsenko, Europhys. Lett. 90, 20003 (2010).
[18] G.Amir,I.Corwin,J.Quastel,Comm.Pur.Appl.Math.
64, 466 (2011).
[19] I. Corwin, Random Matrices: Theory Appl. 1, 1130001
(2012)
[20] K. A. Takeuchi, M. Sano, Phys. Rev. Lett. 104, 230601
(2010);K.A.Takeuchi,M.Sano,T.Sasamoto,H.Spohn,
Sci.Rep.(Nature)1,34(2011);K.A.Takeuchi,M.Sano,
J. Stat. Phys. 147, 853 (2012).
[21] L.Miettinen,M.Myllys,J.Merikosks,J.Timonen,Eur.
Phys. J. B 46, 55 (2005).
[22] ForareviewofrecentadvancesintheKPZproblem,see
T.Halpin-Healy,K.A.Takeuchi,J.Stat.Phys.160,794
(2015).
[23] The heights at the two edge points x = ±t, h (t) =
±
h(x = ±t,t), evolve differently: h (t) = h (t−1)+
± ±
η(±t,t).
[24] P.Vivo,S.N.Majumdar,O.Bohigas,J.Phys.A:Math.
6
.
SUPPLEMENTARY MATERIAL
JOHANSSON MODEL
Johansson’s directed polymer model in 2-dimensions [10] is defined as follows. Consider a 2-d lattice where a site
(i,j) has a quenched energy η(i,j), drawn independently for each site from an exponential distribution: p(η) = e η
−
withη 0. Considernowadirectedpathfromtheorigintothesite(M,N)(M 0,N 0). Theenergyofapathis
≥ ≥ ≥
justthesumoftheenergiesofallsitesbelongingtothepath. Fromallpossiblepathsendingat(M,N),oneconsiders
the optimal path, i.e., the one with the highest energy. Let E(M,N) denote the energy of this optimal path. One
can easily write a recursion relation
E(M,N)=max[E(M 1,N),E(M,N 1)]+η(M,N). (29)
− −
Clearly, E(M,N) is a random variable and one is interested in its probability distribution. Making the change of
variables, x = M N and t = M +N and denoting E(M,N) h(x,t), it reduces to an interface growth model,
− ≡
where the height h(x,t) ( t x t), evolves with discrete time t according to the following rules (see Fig. 2),
− ≤ ≤
h(x,t)=max[h(x 1,t 1),h(x+1,t 1)]+η(x,t) for t<x<t. (30)
− − − −
At the two edge points x= t, the evolution of the height h( t,t) h (t) is slightly different
± ± ≡ ±
h (t)=h (t 1)+η(t,t) (31)
+ +
−
h (t)=h (t 1)+η( t,t). (32)
− − − −
At late times, the average height at point x converges to [10]
(cid:16)x(cid:17)
h(x,t) v t; t x t (33)
(cid:104) (cid:105)→ t − ≤ ≤
where v(z = x/t) = 1+√1 z2 has a semi-circular form (see Fig. 2). The height h(x,t) fluctuates around this
−
average typically on a scale (t1/3) for large t. In particular, at x = 0, the height at late times converges to
∼ O
h(0,t) 2t+2t1/3χ , where the random variable χ is of (1) (independent of t for large t) and is distributed via
2 2
→ O
the Tracy-Widom GUE law [10]. In other words, the PDF of the scaled (and centered) height at the origin
h(0,t)
H = t (34)
2 −
has the late time scaling form
(cid:18) (cid:19)
1 H
P(H,t) f (35)
∼ t1/3 2 t1/3
where f (s) is the TW GUE PDF with asymptotics given in Eq. (2) of the main text with β =2. This is represented
2
schematically by the central blue region in Fig. 1 of the main text.
Incontrasttothetypicalfluctuations,theatypicallylargefluctuationsbothtotheleftandtotherightofthemean,
are not described by the Tracy-Widom distribution. To compute these tails, one can use an exact mapping due to
Johansson [10] that states
Prob.[E(M,N) l]=Prob.[λ l], (36)
max
≤ ≤
where λ denotes the largest eigenvalue of an (M N) complex Wishart matrix defined as follows. Let X be an
max
×
(M N)rectangularmatrixwhoseentriesareindependentcomplexGaussianvariables, Prob.[X] exp[ Tr(X X)].
†
× ∝ −
7
ConstructthentheproductmatrixW =X X whichis(N N)andhasN non-negativerealeigenvaluesλ ,λ ,...,λ
† 1 2 N
×
with maximal eigenvalue λ =max(λ ,λ ,...,λ ). Without any loss of generality, one can assume N M. The
max 1 2 N
≤
statistics of λ has been studied extensively in the random matrix literature and one can then borrow these results
max
for our problem. In terms of the height, the relation (36) simply reads
Prob.[h(x,t) l]=Prob.[λ l], (37)
max
≤ ≤
where x = M N and t = M +N. Since we are interested in the height at x = 0, this corresponds to the Wishart
−
matrix with M = N and N = t/2. For M = N, it is well known [10] that for large N, λ 4N +24/3N1/3χ .
max 2
→
Using N = t/2, one immediately recovers the result that h(0,t) 2t+2t1/3χ for large t as mentioned above. In
2
→
addition, the large deviation tails of λ for Wishart matrices are also known [24, 25]. For M = N, they read as
max
N
→∞
(cid:20) (cid:18) (cid:19)(cid:21)
l l
Prob.[λ l] exp N2ψW for 0 4 (38)
max
≤ ∼ − − N ≤ N ≤
(cid:20) (cid:18) (cid:19)(cid:21)
l l
1 exp NψW for 4 (39)
∼ − − + N N ≥
where the left rate function ψW(y) is given explicitly as [24]
−
(cid:16) y(cid:17) 1(cid:16) y(cid:17)2
ψW(y)=ln4 lny 1 1 for 0 y 4, (40)
− − − − 4 − 2 − 4 ≤ ≤
while the right rate function ψW(y) has the expression [25]
+
(cid:112) (cid:16) (cid:112) (cid:17)
ψW(y)= ln4+ y(y 4)+2 ln y 2 y(y 4 for y 4. (41)
+ − − − − − ≥
Note that the superscript W stands for Wishart matrices.
To translate these results to the height model and derive the large deviation results mentioned in Eq. (1) in the
main text, we consider the scaled height defined in Eq. (34). Then, using N =t/2, we get
Prob.[H tz]=Prob.[h(0,t) 2(1+z)t]=Prob.[λ 4(1+z)N]. (42)
max
≤ ≤ ≤
Finally, using the results from Eqs. (38) and (39) and using again N =t/2 we obtain the announced results
Prob.[H tz] exp(cid:2) t2Φ (z)(cid:3) for 1 z 0 (43)
≤ ∼ − − − ≤ ≤
1 exp[ tΦ (z)] for z 0 (44)
+
∼ − − ≥
where the rate functions Φ (z) can be expressed explicitly in terms of the Wishart rate functions in Eqs. (40) and
±
(41). We get
1 1 (cid:20) z2 (cid:21)
Φ (z)= ψW(4(1+z))= z ln(1+z) for 1 z 0, (45)
− 4 − 4 − 2 − − ≤ ≤
1 (cid:112) (cid:16) (cid:112) (cid:17)
Φ (z)= ψW(4(1+z))=2 z(1+z)+ln 2z+1 2 z(1+z) for z 0. (46)
+ 2 + − ≥
Taking derivatives with respect to z in Eqs. (43) and (44), one gets the large deviation tails of the PDF P(H,t) of
the scaled height H at the origin as announced in Eqs. (5) and (6) respectively in the main text.
Note that while the large deviation principle in this problem was originally established by Johansson [10], the left
rate function Φ (z) was not computed. Here we obtain this function explicitly in (45). While a general expression
−
for the right rate function Φ (z) was computed by Johansson for the geometric disorder, here we obtain a simplified
+
explicit expression for Φ (z) in (46) for the exponential disorder.
+
Matching with the tails of the Tracy-Widom distribution:
We start from the left tail. When the scaled height z =H/t in Eq. (34) approaches 0 from below, it is easy to see
by expanding Φ (z) to leading order for small z
−
z 3
Φ (z) | | . (47)
− ∼ 12
8
Substituting this result in Eq. (43) and taking a derivative with respect to z, one finds that when z 0 , the left
−
→
large deviation tail of the PDF of H behaves as
(cid:20) H 3(cid:21)
P(H,t) exp | | . (48)
∼ − 12t
Ontheotherhand,ifwestartfromthecentralTracy-Widomdistributionthatdescribestypicalfluctuationsof (t1/3)
O
in Eq. (35), and set H = zt, we will probe the probability of fluctuations to the left that are much larger (of (t))
O
than the typical size (t1/3). This gives
O
(cid:16) (cid:17)
P(H =zt,t) t 1/3f zt2/3 . (49)
− 2
∼
As t with fixed z <0, the argument of f in Eq. (49) tends to negative infinity. So, we need to use the left tail
2
→∞
asymptotic of the Tracy-Widom density in Eq. (2) of the main text: f (s) exp[ s3/12]. Substituting this in Eq.
2
∼ −| |
(49) gives P(H,t) exp[ H 3/12t], which matches smoothly with the result in Eq. (48) obtained from the small
∼ −| |
argument behavior of the left large deviation regime.
Asimilarmatchingcanbeverifiedontherightsideaswell. Whenz approaches0+ fromabove,wegetbyexpanding
Φ (z) to leading order
+
4
Φ (z) z3/2. (50)
+
∼ 3
Substituting in Eq. (44) and taking a derivative with respect to z, one finds that when z 0+, the right large
→
deviation tail of the PDF of H behaves as
(cid:20) 4 H3/2(cid:21)
P(H,t) exp . (51)
∼ −3 √t
In contrast, starting from the central TW regime (valid on a scale H t1/3), and setting H = zt gives Eq. (49)
∼
wherez >0. Ast withfixedz >0, theargumentoff inEq. (49)nowtendstopositiveinfinity. Hence, weuse
2
→∞
the right tail asymptotic of the Tracy-Widom density in Eq. (2) of the main text: f (s) exp[ 4s3/2]. Substituting
2 ∼ −3
this in Eq. (49) gives P(H,t) exp[ 4 H3/2], which then matches smoothly with the result in Eq. (51) obtained
∼ −3 √t
from the small argument behavior of the right large deviation regime.
Note that although here we have restricted ourselves, for simplicity, to the height at the origin x = 0, the compu-
tations presented above can be easily extended to the large deviations of the height h(x,t) at a generic point x.
RIGHT TAIL ASYMPTOTICS OF THE KERNEL AT EQUAL POINTS
In Eq. (17) of the main text, for the simplicity of reading, we only provided the leading exponential factor for the
asymptotic expansion of the kernel. However, one can easily obtain also the subdominant pre-exponential factors as
shown below.
We start by evaluating the asymptotic behavior of the integral on the r.h.s. of (17) with y = (1) fixed and as
O
t . It turns out that the dominant contribution to this integral comes from the interval v [ ,y]. In this
→ ∞ ∈ −∞
interval, for large t, we can replace the Airy function by its large positive tail asymptotics Ai(z)(cid:39) √4π1z1/2e−23z3/2 as
z + . This leads to
→ ∞
K (yt2/3,yt2/3) t1/3 (cid:90) y dv e−t43(y−v)3/2 . (52)
t (cid:39) 4π (y v)1/2 1+evt
−∞ −
It turns out that there are two regimes (i) y >1/4 (ii) 0<y <1/4.
In the first regime y > 1/4, the integral can be evaluated by the saddle point method. We first assume, and then
check a posteriori, that there is a saddle point v > 0. Then the integral will be dominated near v > 0. Then, one
∗ ∗
can replace 1/(1+evt) by e vt for large t with v >0 and evaluate the integral by the saddle point method:
−
t1/3 (cid:90) y dv
K (yt2/3,yt2/3) e tS(y,v) (53)
t (cid:39) 4π (y v)1/2 −
−∞ −
4
S(y,v)= (y v)3/2+v . (54)
3 −
9
For y >1/4, the saddle point is at v =y 1. For consistency we need v >0, i.e., y >1/4. Evaluating the integral
∗ − 4 ∗
at this saddle point gives
1
Kt(yt2/3,yt2/3) e−t(y−112) . (55)
(cid:39) √4πt1/3
In the second regime 0 < y < 1/4, there is no saddle point and the dominant contribution to the integral in (52)
comes from the edge v 0. Setting v =w/t and keeping only leading order terms for large t we obtain
≈
K (yt2/3,yt2/3) e−t43y3/2 (cid:90) +∞dwe2√yw (56)
t (cid:39) 4πt2/3√y 1+ew
−∞
This integral can be performed explicitly giving
Kt(yt2/3, yt2/3)(cid:39) 4t2/3√ey−ts34iny3(cid:0)/22π√y(cid:1). (57)
If we neglect the pre-exponential factors we recover the formula given in the text, namely
K (yt2/3,yt2/3) e tI(y) (58)
t −
∼
(cid:40)
4y3/2 , 0<y < 1
I(y)= 3 4 (59)
y 1 , y > 1 .
− 12 4
PRE-EXPONENTIAL FACTOR IN THE RIGHT LARGE DEVIATION TAIL
Inspired by the form of the subdominant corrections in the right large deviation tail of the top eigenvalue of a
Gaussian random matrix [36], it is natural to make the following ansatz in the limit of large time
4
lnP(H,t) t z3/2 alnt χ(z)+o(1) , z =H/t fixed. (60)
(cid:39)− 3 − −
In this section we establish this behavior, both using moments from the replica method and using the exact form of
the generating function. We also calculate a and χ(z) explicitly, both for the flat as well as droplet initial conditions
and show that they do depend on the initial conditions.
Moments from the replica method
The positive integer moments of eH for the continuum KPZ equation can be studied using the mapping to the
attractive Lieb-Liniger model with n bosons [32]. From the Bethe ansatz solution of this model the exact formula for
the moments at arbitrary time [16, 17] takes the form of a sum of exponentials
(cid:88)
(cid:104)enH(cid:105)= Bµn,t e−Eµnt (61)
µn
where the index µ labels the n- boson eigenstates. In the limit of large system size L = + , these are made of
n
∞
so-called strings, with a total energy spectrum
(cid:88)ns 1 (cid:88)ns
E = m k2 m3 , m =1 , m 1 , 1 n n (62)
µn j j − 12 j j j ≥ ≤ s ≤
j=1 j=1
where the k are the (real) momenta of each string.
j
At large time and fixed positive integer n, the sum (61) is dominated by the ground state 0 , together with
n,k=0
| (cid:105)
its center of mass finite momentum excitation, i.e. more precisely, taking into account the gap with the next set of
excited states
enH t + An,te112n3t[1+ (e−41n(n−1)t)] (63)
(cid:104) (cid:105)∼ → ∞ O
10
where the amplitude (see e.g. [37])
A = lim 1 (cid:88) e nk2t 1 Ψ 0 . (64)
n,t L + L − n2(cid:104) 0| (cid:105)n,k
→ ∞ k=2πp,p Z
nL ∈
The last factor is the overlap, i.e., the scalar product of the (unnormalized) ground state wave function (such that
0,..00 = n!), with the (unnormalized) replica wave function Ψ encoding for the initial condition. This overlap
n 0
(cid:104) | (cid:105) | (cid:105)
is complicated in general, but is known for some special initial conditions. This leads for n 1 to
≥
Ψ 0 =n22n 1L δ A =2n 1 , flat initial condition, (65)
0 n,k − k,0 n,t −
(cid:104) | (cid:105) ⇒
n!
Ψ 0 =n! A = , droplet initial condition. (66)
(cid:104) 0| (cid:105)n,k ⇒ n,t n3/2(4πt)1/2
The saddle point method described in the text can be extended to obtain the pre-exponential factor. Substituting
the anticipated form (60) we obtain for any fixed integer n>0 and large t
(cid:90)
enH t dze−t(34z3/2−nz)−alnt−χ(z) t12−a√πn et1n23−χ(n2/4) , (67)
(cid:104) (cid:105)(cid:39) (cid:39)
obtained using the saddle point at z =z =n2/4.
n
In the flat initial condition case, comparing (63), (65) with (67) one finds a = 1/2 and the correction to scaling
function
1 1
χ (z)= ln(8π)+ lnz (ln4)√z , z =z =n2/4 , n N . (68)
flat n ∗
2 4 − ∈
In the droplet case we get a=1 and χ (n2/4)=ln(2πn2/n!), hence the correction to scaling function
droplet
χ (z)=ln(4π)+ 1lnz ln(cid:0)Γ(2√z)(cid:1) , z =z =n2/4 , n N . (69)
droplet n ∗
2 − ∈
From moments to the generating function
Expanding the generating function in Eq. (10) in terms of moments, reads
gt(s)=1+(cid:88) (−1)ne−nt1/3s enH 1 (cid:90) ∞ duAi(2u+22/3s)(1 e−2e21/3t1/3u) , flat initial condition(70)
n! (cid:104) (cid:105)(cid:39) − −
n≥1 −∞
(cid:90) ∞ (cid:90) ∞ Ai(r+u)2
1 dr du , droplet initial condition. (71)
(cid:39) − 1+e t1/3u
s −
−∞
Toobtainthefirstlineweused(63),(65)andthe”Airytrick”identity(cid:82)+∞dyAi(y)eyw =ew3/3 forw >0. Toobtain
the second line we used (63), (66) and the following variant −∞
e nt1/3s en3t/12 =(cid:90) +∞dr(cid:90) ∞ duAi(r+u)2ent1/3u (72)
−
n3/2(4πt)1/2
s
−∞
for n>0, and then summed up the geometric series in n (see [16] and Section 4.2.1 in [37] for details). In the droplet
case it recovers the expansion (15) and for t + in the flat case it also reproduces (15) where K (r,r ) is replaced
t (cid:48)
→ ∞
by the GOE kernel Ai(r+r ). The asymptotics of these kernels then allow to recover the asymptotics obtained by
(cid:48)
thesaddlepointmethod, showingthat, toobtaintherighttaillargedeviations, itisequivalenttoworkonthereplica
formula or on the generating function, as mentioned in the text and also done below.
Right tail from the generating function: droplet initial condition
Taking a derivative of g (s) with respect to s in (10) we obtain the relation (valid for all t and large s)
t
(cid:104)eH−st1/3−eH−st1/3(cid:105)= t11/3Kt(s,s). (73)