Universality of the Pearcey process

0.48 MB·English

by  Mark Adler

Universality of the Pearcey process
9 Mark Adler∗ Nicolas Orantin† Pierre van Moerbeke‡
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Contents
8
2
1 Introduction 2
]
R
P 2 Non-intersecting Brownian motions on R, forced to several
h. points 9
t
a
m 3 Proof of Theorem 1.1 12
[
4 Steepest descent analysis 20
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v
0 5 Proof of Theorem 1.2 25
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5 ∗2000 Mathematics Subject Classification. Primary: 60J60, 60J65, 60G55; secondary:
4 35Q53, 35Q58. Key words and Phrases: Non-intersecting Brownian motions, Pearcey
.
1 distribution,matrixmodels,randomHermitianensembles,multi-componentKPequation,
0 Virasoro constraints.
9 Department of Mathematics, Brandeis University, Waltham, MA 02454, USA. E-mail:
0
[email protected]. The support of a National Science Foundation grant # DMS-04-
:
v 06287 is gratefully acknowledged.
Xi †Universit´e de Louvain, 1348 Louvain-la-Neuve, Belgium. E-mail:
[email protected]. The support of the ENRAGE European network
r
a MRTN-CT-2004-005616, the ENIGMA European network MRT-CT-2004-5652, the
French and Japanese governments through PAI SAKURA, the European Science Foun-
dation through the MISGAM program and the ANR project G´eom´etrie Int´egrabilit´e en
Physique Math´ematique ANR-BLAN-0029-01 is gratefully acknowledged.
‡Department of Mathematics, Universit´e de Louvain, 1348 Louvain-la-Neuve, Bel-
gium and Brandeis University, Waltham, MA 02454, USA. E-mail: vanmoer-
[email protected] and @brandeis.edu. The support of a National Science Foundation
grant#DMS-04-06287, aEuropeanScienceFoundationgrant(MISGAM),aMarieCurie
Grant (ENIGMA), Nato, FNRS and Francqui Foundation grants is gratefully acknowl-
edged.
1
6 An estimate for the Wronskian 30
7 Steepest descent analysis and replica duality 35
7.1 From Hermitian matrix integrals to double contour integrals . 35
7.2 Saddle points, spectral curve and universality . . . . . . . . . 39
7.3 Steepest descent analysis . . . . . . . . . . . . . . . . . . . . . 40
7.3.1 Simple branch point: the Airy kernel . . . . . . . . . . 41
7.3.2 Double branch point: the Pearcey kernel . . . . . . . . 42
7.3.3 kth order branch point . . . . . . . . . . . . . . . . . . 43
7.4 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.5 Example: backtothematrixmodelandnon-intersectingBrow-
nian motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.6 Application: Brownian bridges from one point to k points . . . 48
Abstract
Consider non-intersecting Brownian motions on the line leaving
from the origin and forced to two arbitrary points. Letting the num-
ber of Brownian particles tend to infinity, and upon rescaling, there
is a point of bifurcation, where the support of the density of parti-
cles goes from one interval to two intervals. In this paper, we show
that at that very point of bifurcation a cusp appears, near which the
Brownian paths fluctuate like the Pearcey process. This is a univer-
sality result within this class of problems. Tracy and Widom obtained
such a result in the symmetric case, when the two target points are
symmetric with regard to the origin. This asymmetry enabled us to
improve considerably a result concerning the non-linear partial differ-
ential equations governing the transition probabilities for the Pearcey
process, obtained by Adler and van Moerbeke.
1 Introduction
Consider the probability that n non-intersecting (Dyson) Brownian motions
x (t) < ... < x (t)
1 n
in R belong to a set E ∈ R, with all particles leaving from the origin at time
t = 0 and all forced to end up at b < b < ... < b at time t = 1:
1 2 p
2
 
all x (0) = 0
j
 all x (t) ∈ E for 1 ≤ j ≤ n n paths end up at b at t = 1 
P(b1,...,bp) j 1 . 1 
n  . 
.
 
n paths end up at b at t = 1
p p
with (cid:80)p n = n and with (local) transition probability
i=1 i
p(t;x,y) := √1 e−(x−y)2. (1.1)
t
πt
A formula by Karlin-McGregor enables one to express this probability as an
integral of a product of two determinants involving the transition probability
(1.1) above. This further leads to a expression as (i) a GUE-matrix integral
withanexternalpotential, (ii)adeterminantofablockmomentmatrix, with
p blocks and (iii) a Fredholm determinant of a kernel. Finally it is also the
solution of a PDE in the end-points of the interval E and the target points
b ,...,b .
1 p
Throughout this paper, we shall be dealing with the case of two target
points p = 2. In this paper, we show that, when n → ∞ and when one looks
throughamicroscopenearacertainpointofbifurcation, thenon-intersecting
Brownian motions tend to a new process, the Pearcey process, whatever be
the location of the target points and whatever be the proportion of particles
forced to those points. Tracy and Widom [28] showed this result in the
symmetric case; namely when the target points are symmetric with respect
to the origin and half of the particles go to either target point. Br´ezin and
Hikami [9, 10, 11, 12] first considered this kernel and Bleher-Kuijlaars [8]
obtained strong asymptotics using Riemann-Hilbert techniques.
ThePearcey processP(t)describesacloudofBrownianparticles,evolving
in time according to a (matrix) Fredholm determinant,
(cid:18) (cid:19)
(cid:16) (cid:17)
PP (cid:0)all P(t ) ∈ Ec,1 ≤ j ≤ m(cid:1) = det I − χ KP χ
j j Ei titj Ej 1≤i,j≤m
of the Pearcey kernel
KP (x,y) = − 1 (cid:90) dV (cid:90) i∞ dUe−U4+tU2−UyeV4−sV2+Vx 1
4 2 4 2
s,t 4π2 U −V
X −i∞
−(cid:112)I(s < t) e−(2x(−t−ys)2)
2π(t−s)
3
(1.2)
The contour X is given by the ingoing rays from ±∞eiπ/4 to 0 and the
outgoing rays from 0 to ±∞e−iπ/4, i.e., X stands for the contour, all rays
making an angle of π/4 with the horizontal axis.
(cid:45) (cid:46)
0
(cid:37) (cid:38)
For s = t, the Pearcey kernel can also be written
p(x)q(cid:48)(cid:48)(y)−p(cid:48)(x)q(cid:48)(y)+p(cid:48)(cid:48)(x)q(y)−tp(x)q(y)
KP(x,y) = (1.3)
t,t x−y
with
q(y) := i (cid:90) i∞ e−U4+tU2−UydU, p(x) := 1 (cid:90) eV4−tV2+VxdV,
4 2 4 2
2π 2πi
−i∞ X
satisfying both, the differential equations (using integration by parts)
p(cid:48)(cid:48)(cid:48)(x)−tp(cid:48)(x)+xp(x) = 0, q(cid:48)(cid:48)(cid:48)(y)−tq(cid:48)(y)−yq(y) = 0, (1.4)
and the heat equations
∂p 1 ∂q 1
= − p(cid:48)(cid:48)(x), = q(cid:48)(cid:48)(y), (1.5)
∂t 2 ∂t 2
whereas KP satisfies the following equation
t,t
∂KP 1
t,t = (−p(cid:48)(x)q(y)+p(x)q(cid:48)(y)). (1.6)
∂t 2
The latter follows from taking ∂/∂t of the kernel (1.2), which has for effect to
multiplytheexponentialsundertheintegral(1.2)with 1(U2−V2)/(U−V) =
2
1(U +V).
2
Considernnon-intersectingBrownianmotions, with0 < p < 1andb < a:
 
all x (0) = 0
j
(cid:92)
P(b,a) {all xj(ti) for 1 ≤ j ≤ n} pn paths end up at a at t = 1 
n
1≤i≤m 
(1−p)n paths end up at b at t = 1
4
When n → ∞, the mean density of Brownian particles has its support on one
interval for t ∼ 0 and on two intervals for t ∼ 1, so that a bifurcation appears
for some intermediate time t , where one interval splits into two intervals. At
0
this point the boundary of the support of the mean density has a cusp. We
show that near this cusp, the same Pearcey process appears, independently
of the values of a, b and p, showing “universality” of the Pearcey process; see
Figure 1. As it turns out, it is convenient to introduce the parametrization
1 (cid:112)
p = with 0 < q < ∞ and let r := q2 −q +1. (1.7)
1+q3
Theorem 1.1 For n → ∞, the cloud of Brownian particles lie within a
√
region, having a cusp at location (x n,t ), with
0 0
(2a−b)q +(2b−a) 1 (cid:18)r(a−b)(cid:19)2
x = t , = 1+2 . (1.8)
0 0
q +1 t q +1
0
Moreover, the following probability tends to the probability for the Pearcey
process:
(cid:32) (cid:33)
(cid:26) (cid:18) (cid:19) (cid:27)
√ √ (cid:92) (cid:16)c µ(cid:17)2 c µ
lim P(b n,a n) all x t + 0 2τ ∈ x n1/2 +c Aτ + 0 Ec
n→∞ n j 0 n1/4 i 0 0 i n1/4
1≤i≤m
(cid:32) (cid:33)
(cid:92)
= PP {P(τ )∩E = ∅} ,
i
1≤i≤m
(1.9)
using the following constants
(cid:18)q2 −q +1(cid:19)1/4 (cid:114)t (1−t ) r(a−b)
0 0
µ = > 0, , c := = t > 0.
0 0
q 2 q +1
q1/2(a−x )+q−1/2(b−x )
0 0
A = (1.10)
(a−b)
In [4], Adler and van Moerbeke showed that the Pearcey transition prob-
ability (1.11) satisfies a non-linear PDE, expressible as a Wronskian of the
expression (1.12) with some partial. This was obtained from taking a scal-
ing limit, when n → ∞, of the symmetric situation, i.e., where b = −a
5
and p = 1/2. It came as a surprise to us that considering the asymmet-
ric case leads to a different non-linear PDE, when n → ∞, but nevertheless
alsoexpressibleasaWronskianofthesameexpression(1.12)withsomeother
partial. A separate functional-theoretical argument then enables one to show
that the expresssion (1.12) itself vanishes. This was one of the motivations
for finding the exact scaling as presented in Theorem 1.1.
To E = ∪r (y ,y ) ⊂ R one associates two operators, a divergence
i=1 2i−1 2i
and an Euler operator
2r 2r
(cid:88) ∂ (cid:88) ∂
∂ = , ε = y .
E ∂y E i∂y
i i
1 1
6
parametrization:
1−p (cid:112)
q3 := , r := q2 −q +1.
p
cusp x−x = 2(cid:0)t−t0(cid:1)3/2 at
0 3
2q −1 1 (cid:18) ar (cid:19)2
x = at , = 1+2 .
0 0
q +1 t q +1
0
Figure 1: The Pearcey process for b = 0.
Theorem 1.2 The log of the transition probability for the Pearcey process,
which is non-stationary,
Q(t,E) := logPP (P(t)∩E = ∅) (1.11)
satisfies the following 3rd order non-linear PDE in t and the boundary points
of E,
∂3Q 1 (cid:18) ∂ (cid:19) 1 (cid:26) ∂Q(cid:27)
+ ε −2t −2 ∂2Q− ∂2Q,∂ = 0, (1.12)
∂t3 8 E ∂t E 2 E E ∂t
∂
E
7
with “final condition”, given by the Airy process1 which is a stationary pro-
cess, (by moving far out along the cusp x = 2(cid:0)t(cid:1)3/2)
3
(cid:32)P(t)−2(cid:0)t(cid:1)3/2 (cid:33)
lim PP 3 ∩(−E) = ∅ = det(I −A)
t→∞ (3t)1/6 (−E)
Remark: It is interesting to compare the Pearcey PDE with the Airy process
PDE; namely for semi-infinite intervals E and E , the 3rd order non-linear
1 2
PDE for the Airy joint probability
(cid:18) (cid:19)
y +x y −x
QA(t;x,y) := logPA A(t ) ≤ , A(t ) ≤ , for t = t −t ,
1 2 2 1
2 2
reads
∂3QA (cid:18) ∂ ∂ (cid:19)(cid:18)∂2QA ∂2QA(cid:19) (cid:26)∂2QA ∂2QA(cid:27)
2t = t2 −x − +8 , ,
∂t∂x∂y ∂x ∂y ∂x2 ∂y2 ∂x∂y ∂y2
y
(1.13)
with ”final condition”:
lim PA(A(t ) ≤ u , A(t ) ≤ u ) = F(u )F(u ).
1 1 2 2 1 2
t2−t1→∞
Inthelastsection(section7), wedevelop-inaformalway-thecentralrole
played by the spectral curve (or Pastur equation [24]) in the steepest descent
analysis used to prove the universal behavior of the kernel as N → ∞ for
the different problems of non-intersecting Brownian motions. The spectral
curve is precisely the function which appears in the steepest descent analysis.
The spectral curve associated to the problem provides the universal limiting
kernel obtained after a proper rescaling of the variable around a singularity
of the problem.
1The Airy process is a stationary process, which describe the statistical fluctuations
of the process about the curve appearing in Figure 1, away from the edge and properly
rescaled. Its probability given at any time by the Tracy-Widom distribution F(x). The
latterisgivenbytheFredholmdeterminantdet(I−A)oftheAirykernelA, restrictedto
the interval under consideration.
8
R
2 Non-intersecting Brownian motions on ,
forced to several points
In the expression below, H (E) is the set of all Hermitian matrices with
n
all eigenvalues in E. Note that in general one has the following, using the
Karlin-McGregor formula2 (see [21, 9, 10, 11, 12, 28, 8]):
 
all x (0) = 0
j
 all x (t) ∈ E for 1 ≤ j ≤ n n paths end up at b at t = 1 
P(b1,...,bp) j 1 . 1 
n  . 
.
 
n paths end up at b at t = 1
p p
(cid:90) (cid:81)ndx
= lim 1 i det(p(t;γ ,x )) det(p(1−t;x ,δ )) ,
Z i j 1≤i,j≤n i(cid:48) j(cid:48) 1≤i(cid:48),j(cid:48)≤n
all γi→0 En n
δ1,...,δn1 →b1
.
.
.
δn1+...+np−1+1,...,δn→bp
(cid:12)
= Z1n (cid:90)E˜n∆n(x1,...,xn)(cid:89)(cid:96)=p1∆n(cid:96)(x((cid:96)))(cid:89)jn=(cid:96)1e−12xj((cid:96))2+˜b(cid:96)x(j(cid:96))dx(j(cid:96))(cid:12)(cid:12)(cid:12)(cid:12) E˜ = E(cid:113)t(12−t)
(cid:113)
˜b = 2t b
(cid:96) 1−t (cid:96)
(cid:90)
1
= dMe−12Tr(M2−2AtM)dM
Z “ q ”
n Hn E t(12−t)
 (cid:18)(cid:90) (cid:19) 
xi+je−x22+˜b1xdx
1  E˜ . 0≤i≤n1−1, 0≤j≤n−1 
= det .. 
Z  
n  (cid:18)(cid:90) (cid:19) 
 xi+je−x22+˜bpxdx 
E˜ 0≤i≤np−1, 0≤j≤n−1
= det(I −H(p)) , (Fredholm determinant) (2.1)
n Ec
2∆ (x ,...,x ) is the Vandermonde determinant.
n 1 n
9
where H(p)(x,y) is the kernel (setting t = t = t)
n k (cid:96)
H(p)(x,y)dy
n
dy (cid:90) (cid:90) e−t1k−Vtk2+12−xVtk (cid:89)p (cid:18)U −br(cid:19)nr 1
= − dV dU
2π2(cid:112)(1−tk)(1−t(cid:96)) C L+iR e−t1(cid:96)−Ut2(cid:96)+12−yUt(cid:96) r=1 V −br U −V

0 for t ≥ t
 k (cid:96)


− . (2.2)
 √ 1 e−(tx(cid:96)−−yt)k2e1−x2tk−1−y2t(cid:96), for tk < t(cid:96)
π(t −t )
(cid:96) k
where X is a contour consisting of the two incoming rays from ±∞eiπ/4 to 0
and the two outgoing rays from 0 to ±∞e−iπ/4, provided no b = 0. In the
r
expression above, A is the diagonal matrix
t
 
˜
b
1
 ... O  (cid:108) n
 ˜  1
 b 
1
 
˜
 b 
 2 
 O ...  (cid:108) n2 ˜ (cid:114) 2t
A :=   . with b = b ,
t  ˜b  .. i i 1−t
2
 
 ... 
 
 ˜ 
 b 
p
 ...  (cid:108) np
 
˜
b
p
(2.3)
The main expression appearing in (2.1) contains the matrix integral
Pn(E;b1,...,bp) = Z1 (cid:90) ∆n(x1,...,xn)(cid:89)p ∆n(cid:96)(x((cid:96)))(cid:89)n(cid:96) e−12xj((cid:96))2+b(cid:96)x(j(cid:96))dx(j(cid:96))
n En
(cid:96)=1 j=1
(cid:90)
1
= dMe−1Tr(M2−2AM), (2.4)
2
Z
n Hn(E)
which is now viewed as a function of the boundary points of E and the target
points b , for which one assumes a linear dependence
i
p p
(cid:88) (cid:88)
c b = 0 with c = 1.
i i i
1 1
10