A PDE for non-intersecting Brownian motions and applications

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A PDE for non-intersecting Brownian motions and applications

Mark Adler¹, Jonathan Del´epine², Pierre van Moerbeke³ and Pol Vanhaecke⁴ Abstract

ConsiderN =n₁+n₂+· · ·+n_p non-intersecting Brownian motions on the real line, starting from the origin at t = 0, with ni particles forced to reachpdistinct target pointsβi at timet= 1, withβ1 < β2 <

· · · < β_p. This can be viewed as a diffusion process in a sector of R^N. This work shows that the transition probability, that is the probability for the particles to pass through windows ˜E_k at times t_k, satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p+ 1, with each row being a derivative of the previous, except for the last column.

It is an interesting open question to understand those equations from a more probabilistic point of view.

As an application of these equations, let the number of particles forced to the extreme pointsβ₁andβ_ptend to infinity; keep the number of particles forced to intermediate points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to −√

n depart from those going to √

n. These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. This work also shows that such equations are an essential tool in obtaining certain asymptotic results. Finally, the paper contains a conjecture.

1Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA, [email protected]. The support of a National Science Foundation grant # DMS-07- 04271 is gratefully acknowledged

2Département de Mathématiques, Université Catholique de Louvain, 1348 Louvain-la- Neuve, Belgium, [email protected]

3Département de Mathématiques, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA, [email protected]. The support of a National Science Foundation grant # DMS-07-04271, a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA), Nato, FNRS and Francqui Foundation grants is gratefully acknowledged.

4The support of a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA) and a ANR Grant (GIMP) is gratefully acknowledged.

arXiv:0911.0152v1 [math.PR] 1 Nov 2009

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1 Introduction

Consider N non-intersecting Brownian motions x₁(t) < x₂(t) < . . . < x_N(t) on R (Dyson’s Brownian motions), all starting at source points γ₁ < γ₂ <

· · · < γ_N at time t = 0 and forced to target points δ₁ < δ₂ < · · · < δ_N at t = 1. According to the Karlin-McGregor formula [17], the probability that theN particles pass through the subsets ˜E₁, E˜₂, . . . , E˜_m ⊂Rrespectively at times 0< t₁ < t₂ <· · ·< t_m <1 is given by (setting t₀ := 0 and t_m+1 := 1),

P

m

\

k=1

n

all x_i(t_k)∈E˜_ko x_j(0) =γ_j, x_j(1) =δ_j, for j = 1, . . . , N

!

= 1 Z_N

Z

E˜₁^N N

Y

i=1

du⁽¹⁾_i Z

E˜₂^N N

Y

i=1

du⁽²⁾_i . . . Z

E˜^N_m N

Y

i=1

du^(m)_i det(p(t₁−t₀;γ_i, u⁽¹⁾_j ))_16i,j6N

×det(p(t₂−t₁;u⁽¹⁾_i , u⁽²⁾_j ))_16i,j6N. . .det(p(t_m+1−t_m;u^(m)_i , δ_j))_16i,j6N (1.1) wherep(t, x, y) denotes the standard Brownian transition probability,

p(t, x, y) := 1

√πt e⁻^(y−x)2^t . (1.2)

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There has been a great deal of interest in non-intersecting Brownian motions and especially in some critical infinite-dimensional diffusions arising when the number of particles N → ∞. This in turn has been motivated by random matrix theory and Dyson’s observation [14] that letting the entries of GUE matrices run according to independent Ornstein-Uhlenbeck processes leads to such non-intersecting Brownian motions for the random eigenvalues of the matrix.

When some source points and some target points coincide, the formula (1.1) for the probability must be adapted by taking appropriate limits; see [17, 16, 10, 7]. In this paper, we consider the situation where the source points all coincide with 0, while some target points may coincide. Consider thusN =n₁+n₂+· · ·+n_p non-intersecting Brownian motions starting from the origin at t = 0, with n_i particles forced to reach p distinct target points βi at time t= 1, with β1 < β2 <· · ·< βp inR; see Figure 1.

Given positive integers n = (n₁, . . . , n_p), given m subsets ˜E₁, . . . ,E˜_m ⊂R and times t₀ = 0< t₁ < t₂ <· · ·< t_m < t_m+1 = 1, this paper deals with the probability⁵P^(β)ⁿ (t,E˜), as in (1.3) below (i.e., the probability for the particles to pass through the windows ˜E_kat timest_k); as is well-known, (see [20, 10, 19, 7]), P^(β)ⁿ (t,E˜) can also be viewed as the probability for the eigenvalues of a chain of m coupled Hermitian random matrices, after some change of variables:

P^(β)n (t,E˜) := P





m

\

k=1

n

all x_i(t_k)∈E˜_ko

all x_i(0) = 0;

n_j paths end up at β_j att= 1, for 16j 6p





= 1

Z˜_n Z

spec(Mk)∈E_k

e⁻

1 2tr^„^P^m

k=1

M_k²−2

m−1

P

k=1

ckMkMk+1−2AMm

« m

Y

k=1

dM_k

=: P^An(c,E). (1.3)

The change of variables is given by the following formulae⁶, A:= diag(

n1

z }| { b₁, . . . , b₁,

n2

z }| { b₂, . . . , b₂, . . . ,

np

z }| {

b_p, . . . , b_p), withb_` = s

2(t_m−tm−1) (1−tm)(1−tm−1)β_` E_k := ˜E_k

s

2(t_k+1−tk−1)

(t_k−tk−1)(t_k+1−t_k), c²_k := (t_k+2−t_k+1)(t_k−tk−1)

(t_k+2−t_k)(t_k+1−tk−1), (1.4) for ` = 1, . . . , p and k = 1, . . . , m. It is quite natural to impose a linear constraint on the rescaled target pointsβ₁, . . . , β_p, namely

p

X

`=1

κ_`β_` = 0, with

p

X

`=1

κ_` = 1, setκ₀ :=−1. (1.5)

5E˜ = ˜E1×. . .×E˜m.

6For m = 1, the matrix integral above becomes a one-matrix integral with external potential. The change of variables below becomes: b`=q

2t

1−tβ`, E= ˜Eq ₂

t(1−t).

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Of course, the same relation holds for theb_i’s. For instance, a typical situation is to take β₁ = −β_p and have all the remaining target points in arbitrary position betweenβ₁ and β_p. This case will be discussed in Section 8.

The natural initial or rather “final condition” for the transition probability (1.3) is given by what happens whent_m →1, keepingt₁, . . . , t_m−1, away from 0 or 1; namely,

tmlim→1P^(β)n (t,E˜) = 0, when ˜E_m ⊃ {β/ ₁, . . . , β_p}. (1.6) It is also known (see ([19])) that the probability aboveP^(β)ⁿ (t₁, . . . , t_m,E˜₁× . . .×E˜_m) = det(1 −χ

Ece i

(x)H_t^(N)

itj(x, y)χ

Ece j

(y)) can be expressed as a matrix Fredholm determinant of a matrix kernel⁷

H_t^(N)_k_,t_`(x, y;β1, . . . , βp)dy = − dy 2π²p

(1−t_k)(1−t_`) Z

C

dV Z

ΓL

dU e⁻^tkV

2 1−tk+₁₋^2xV

tk

e

−t`U2 1−t` +₁₋^2yU

t`

×

p

Y

r=1

U −βr

V −β_r nr

1 U −V

−







0, for t_k >t_`

√ dy

π(t`−tk)e⁻

(x−y)2 t`−tk e ^x

2 1−tk−₁₋^y²

t`, for t_k < t_`

(1.7) whereC is a closed contour enclosing all the points β_r, which is to the left of the line ΓL:=L+iRby pickingLlarge enough, guaranteeing<e(U−V)>0.

These non-intersecting Brownian motions x₁(t) < . . . < x_N(t) describe a diffusion process in a sector {x₁ < x₂ < . . . < x_N} of R^N and thus satisfy a diffusion equation. When the numberN of particles tends to∞, the transition

7The Fredholm determinant of a matrix kernelHbt_it_j(x, y) :=χE_i(x)Ht_it_j(x, y)χE_j(y):

det

I−z(Hbt_it_j)16i,j6m

= 1+

∞

X

n=1

(−z)ⁿ X

06ri6n Pm

1 ri=n

Z

R r1

Y

1

dα⁽¹⁾_i . . .

rm

Y

1

dα^(m)_i det

Hb_t_k_t_`(α^(k)_i , α^(`)_j )

16i6rk 16j6r`

!

16k,`6m

,

where then-fold integral in each term above is taken over the range

R=











−∞< α⁽¹⁾₁ 6. . .6α⁽¹⁾r₁ <∞ ...

−∞< α^(m)6. . .6α^(m)<∞









 .

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Figure 1: Non-intersecting Brownian motions

probability would have to satisfy an “infinite-dimensional diffusion equation”, which however would be very difficult to use. The main result of this paper is to show that this transition probabilityP^An(c,E) satisfies a non-linear PDE in the boundary points of E1, . . . , Em, the target points b1, . . . , bp, and the couplings c₁, . . . , cm−1. It is the determinant of a certain matrix of size p+ 1;

p being the number of target points; so, when the number of particles tends to∞, the form of this equation remains the same, which will be exploited in the limit discussed in Theorem 1.3. Moreover, this determinant misses to be aWronskian by the last column only.

The PDE for the transition probability stems largely from integrable theory; this at least is our approach in the present paper. The integrable theory behind non-intersecting Brownian motions has been developed by us in [8];

the latter contains many different ingredients; among them, multi-component KP hierarchies [18, 6] and multiple-orthogonal polynomials [4, 9, 10]. It is – in our opinion – an interesting open question to understand the PDE from a more probabilistic point of view and to use more conventional probabilistic tools to derive them.

Throughout the paper, we shall use, without further warning, the following notation: (i) The inverse of the following Jacobi matrix will play an important

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role:

J :=







−1 c₁

. .. 0

c₁ −1 . ..

. .. . .. . ..

. .. −1 cm−1

0 . ..

cm−1 −1







−1

. (1.8)

(ii) For any given vectoru= (u₁, . . . , u_α), we denote by

∂_u :=

α

X

i=1

∂

∂u_i, ε_u :=

α

X

i=1

u_i ∂

∂u_i. (1.9)

In particular, given any interval or disjoint union of intervalsE =∪^r_i=1[z2i−1, z_2i], we denote by

∂E :=

sum of partials in the boundary points of E

=P2r i=1

∂

∂zi

ε_E :=

Euler operator in the boundary points of E

=P2r i=1z_i_∂z^∂

i.

(1.10)

(iii) In view of the Theorem below, given b = (b₁, . . . , bp−1) and subsets E_i, define the linear differential operators:

∂_b^(`) :=

p−1

X

i=1

(κ_`−δ_`,i) ∂

∂b_i, ∂_b⁽⁰⁾ := 0, implying

p

X

`=1

∂_b^(`)= 0,

∂_` := ∂_b^(`)−κ_`

m

X

i=1

∂_Ei ×

J_1i for `= 0, J_mi for 16`6p, εb :=

p−1

X

1

bi

∂

∂b_i, ε0 := ε_E

1 −δ1,mε_b−c1

∂

∂c₁, εm :=ε_Em −ε_b−cm−1

∂

∂cm−1

. (1.11) For brevity in the statement of the Theorem, set⁰ :=∂₀ =Pm

1 J_1i∂_Ei.

Theorem 1.1 The probability Pn := P^An(c,E), as in (1.3), with the linear constraint (1.5) on the rescaled target points, satisfies a non-linear PDE in the boundary points of the subsets E1, . . . , Em and in the target points b1, . . . , bp; it is given by the determinant of a(p+ 1)×(p+ 1) matrix, nearly a Wronskian

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for the operator ⁰ :=∂₀,

det







F₁ F₂ F₃ . . . F_p G₀ F₁⁰ F₂⁰ F₃⁰ . . . F_p⁰ G1

F₁⁰⁰ F₂⁰⁰ F₃⁰⁰ . . . F_p⁰⁰ G₂ ... ... ... ... ... F₁^(p) F₂^(p) F₃^(p) . . . Fp^(p) Gp







= 0, (1.12)

where the F_` and G_` are given by F` = −∂0∂`lnPⁿ−n`J1m, G_`+1 := ∂₀G_`+

p

X

i=1

(∂₀)^`F_i ∂₀H_i⁽¹⁾

F_i −∂_iH_i⁽²⁾ F_i

!

, G₀ := 0,

H_`⁽¹⁾ := (κ_`(δ_1,m−ε_m)∂₀+ 2J_1m∂_b^(`)) lnPn+C_`, (1.13) H_`⁽²⁾ := (δ_1,m−ε₀+ 2J_1mb_`∂₀)∂_`lnPn,

with

C`:= 2n`J1m Jmmb`−X

i6=`

ni

b_`−b_i

!

. (1.14)

Thefinal condition(1.6) translates into an “ initial condition” nearcm−1 →0 and b_` → ∞, upon using the fact that

cm−1 '√

1−t_m, cm−1b_` 'O(1).

As a special case, we consider the one-time probability P^(β)ⁿ (t,E) for 0˜ <

t₁ = t < 1. For this case, (1.3) becomes a one-matrix model with external potential Pn := P^An(E), thus with no coupling. The expressions for (1.13) can be replaced by simpler expressions; note that the H_`⁽¹⁾ in (1.15) below are not obtained from the H_`⁽¹⁾, as in (1.13), by setting m = 1; in fact, a further simplification occurs in the equations; also the functions G_` are only specializations of the above G_` up to a sign −(−1)^` and ⁰ now denotes ∂_E instead of∂₀ =−∂_E. In this statement, we use the operator ∂_b^(`) as in (1.11), and we use the following simple operator, in accord with (1.9):

ε :=ε_E −ε_b, with ε_b =

p−1

X

1

b_i ∂

∂b_i.

Corollary 1.2 When m= 1 (the one-time case), then lnPn = lnP^An(E) satisfies the same non-linear PDE (1.12), but with simpler expressions F_` and

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H_`⁽¹⁾ and with ⁰ =∂_E, F_` :=

∂_b^(`)+κ_`∂_E

∂_ElnPn+n_`, H¯_`⁽¹⁾ :=

−κ_`∂_Eε+ (κ_`(ε−1) + 2)(∂_b^(`)+κ_`∂_E)

lnPn+ ¯C_`, H_`⁽²⁾ := (1−ε+ 2b_`∂_E)

∂_b^(`)+κ_`∂_E

lnPn, (1.15)

G_`+1 := ∂_EG_`+

p

X

i=1

(∂_E)^`F_i ∂_EH¯_i⁽¹⁾ Fi

−∂_b⁽ⁱ⁾H_i⁽²⁾ Fi

!

, G₀ = 0, C¯_` := −2n_` (1−κ_`)b_`+X

j6=`

n_j b_`−b_j

! .

In section 7, we shall work out two examples, immediate applications of the equations in Theorem 1.1 and Corollary 1.2. In the first example, we describe nonintersecting Brownian motions, leaving from 0 and forced back to 0. The second example deals with the situation of several target points with the extreme ones being symmetric with regard to the origin. That model will also be used later in Section 8.

Pearcey process with inliers: In section 8, we consider non-intersecting Brownian motions leaving from 0 and forced to ptarget points at time t= 1, with the only condition that the left-most and right-most target points are symmetric with respect to the origin, with p−2 intermediate target points thrown in totally arbitrarily; it is convenient to rename the target points β₁ < . . . < β_p, as follows:

˜

a < −˜c₁ < . . . < −˜cp−2 < −˜a

n₊ n₁ . . . n_p−2 n₋

(1.16) with the corresponding number of particles forced to those points at time t= 1. The purpose of this section is to identify the critical process obtained by letting n := n₊ = n− → ∞ and by rescaling ˜a and the ˜c_i accordingly, while keeping n₁, . . . , n_p−2 fixed. We let ˜a go to −∞ like −√

n and −˜a to

∞ like √

n. The target points −˜c₁, . . . ,−˜cp−2 of the inliers move to ∞ as well, but at a much slower rate, namely like −u_` ⁿ₂1/4

. A new process will appear at the point of bifurcation, where the bulk of the particles forced to

−√

n depart from those going to √

n, namely thePearcey process with inliers, which generalizes the Pearcey process found by C. Tracy and H. Widom [19].

It describes the statistical fluctuations near that point of bifurcation; it will be sensitive to the presence of inliers and will be different in the absence of inliers (Pearcey process). We will compute the kernel governing the transition probabilities and also apply the formulae obtained in Corollary 1.2 to compute a PDE for the gap probability, which, to our surprise, appears to be anexact

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Theorem 1.3 Pick timesτ₁ < . . . < τ_m, subsetsE_j ⊂Rforj = 1, . . . , mand parameters u_` for ` = 1, . . . , p−2. Consider 2n+Pp−2

`=1n_` non-intersecting Brownian motions, such that

(i) all particles leave from 0 at time t = 0, (ii) n =n_± particles are forced to ±√

n at time t = 1, (iii) n_` paths are forced to points⁸ −u_` ⁿ₂1/4

at time t = 1 (16`6p).

Then the following Brownian motion limit holds for the gap probability, about timet = 1/2, keeping n_` fixed,

n→∞lim P

m

\

j=1

all x_i1

2 + τ_j 4√

2n

∈ E_j^c 4(n/2)^1/4

!

= P^P ^(u¹^,...,u^p−2⁾

m

\

j=1

{P(τ_j)∩E_j =∅}

!

= det

1−

χEiK_τ^P_i_τ_jχ

Ej

16i,j6m

, (1.17)

where this probability is given by the Fredholm determinant of the Pearcey matrix kernel with inliers, which is a rational perturbation of the customary Pearcey kernel⁹, namely

K_s,t^P(X, Y; u₁, . . . , u_p−2)

= − 1

4π² Z

X

dV Z i∞

−i∞

dU 1

U −V e⁻^U

4

4 +^tU₂²−U Y

e⁻^V⁴⁴⁺^sV²²^{−V X}

p−2

Y

`=1

U +u_` V +u`

n`

−







0 for t−s60

√ 1

2π(t−s)e⁻

(X−Y)2

2(t−s) for t−s >0. (1.18)

The log of the gap probability (E=E₁× · · · ×E_m) Q(τ1, . . . , τm;u1, . . . , up−2;E) := lnP^P ^(u¹^,...,u^p−2⁾

m

\

j=1

{P(τj)∩Ej =∅}

!

satisfies a partial differential equation, which is ap×pWronskian with respect to the operator ∂_E =Pm

i=1∂_Ei: Wp

∂²

E∂τQ, ∂²

E

∂Q

∂u₁, . . . , ∂²

E

∂Q

∂up−2

, X

∂_E

= 0, (1.19)

8Note that those points belong to the interval [−√ n,√

n] for large enoughn.

9Xstands for the contour

- .0

% &

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Figure 2: Pearcey process with inliers where¹⁰

X:= (ε_E−εu+ 2ετ−2)∂²

EQ+ 4∂u∂_E∂τQ+ 8∂_τ³Q−4 ˜∂_E∂_E∂τQ+ 4

∂_E∂τQ, ∂²

EQ _∂

E

. (1.20) For one-time (m= 1), the expression X reads as follows:

X:= (ε_E−ε_u−2τ ∂

∂τ−2)∂²

EQ+4∂_u∂_E∂Q

∂τ +8∂³Q

∂τ³ +4

∂_E∂Q

∂τ, ∂²

EQ

∂_E

. (1.21) Remark: The term εu∂_E²Q could be omitted in the definition of X, since it is a linear combination of (p−1) columns in the matrix (1.19) (from the second ×u₁ to the (p−1)st column ×up−2). We nevertheless keep this term in the expression, in view of Conjecture 1.5.

In the absence of inliers, one obtains, in particular, the PDE for the transition probability of the Pearcey process: it is a 2 ×2 Wronskian with X as in (1.20) and (1.21), but without the u-partials. In [3], it is shown that the transition probability of the Pearcey process satisfies the simpler equation X= 0.

Corollary 1.4 [3] In the absence of inliers (p= 2), Q(τ₁, . . . , τ_m;E) := lnP^P

m

\

j=1

{P(τ_j)∩E_j =∅}

!

10Remember for u = (u₁, . . . , u_p−2) and τ = (τ₁, . . . , τ_m), one has ∂_u = Pp−2 1

∂

∂ui, εu =Pp−2

1 ui ∂

∂u_i, ∂τ =Pm 1

∂

∂τ_i.One also needs ε_E := Pm

1 εE_i and the mixed time-space

m

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satisfies

(ε_E + 2ε_τ−2)∂²

EQ+ 8∂_τ³Q−4 ˜∂_E∂_E∂_τQ+ 4

∂_E∂_τQ, ∂²

EQ _∂

E

= 0, and for the one-time case (m= 1),

(ε_E −2τ ∂

∂τ −2)∂²

EQ+ 8∂³Q

∂τ³ + 4

∂_E∂Q

∂τ , ∂²

EQ

∂_E

= 0.

We now formulate a conjecture, stating that, even with inliers, the equation for the transition probability reads X = 0, where X is given by (1.20) and (1.21):

Conjecture 1.5 Even with inliers (p >2), we conjecture that the function Q(τ₁, . . . , τ_m;u₁, . . . , up−2;E) := lnP^P ^(u¹^,...,u^p−2⁾

m

\

j=1

{P(τ_j)∩E_j =∅}

!

satisfies

X= (ε_E−ε_u+2ε_τ−2)∂²

EQ+4∂_u∂_E∂_τQ+8∂_τ³Q−4 ˜∂_E∂_E∂_τQ+4

∂_E∂_τQ, ∂²

EQ _∂

E

= 0, (1.22) and for the one-time case (m= 1),

X= (ε_E −ε_u−2τ ∂

∂τ −2)∂_E²Q+ 4∂_u∂_E∂Q

∂τ + 8∂³Q

∂τ³ + 4

∂_E∂Q

∂τ , ∂_E²Q

∂_E

= 0.

(1.23)

The PDE’s play a prominent role in obtaining certain approximations which would be very hard to obtain without that technology. An example will be given here, without proof, for the Pearcey process without inliers. At the point of bifurcation, mentioned above, there appears a cusp in the Pearcey scaleξ =±₂₇² (3τ)^3/2, such that, roughly speaking, most Pearcey process paths stay completely to the left or to the right of this cusp. Upon comparing the Pearcey process with, say, the right branch of the cusp in the new (crude) space-scale (3τ)^1/6, and letting two different times τ1 and τ2 tend to ∞ in a very specific way, one is led to the so-called Airy process A(t). The exact approximation is given in the Theorem below taken from [1]:

Theorem 1.6 Let τ₁, τ₂ → ∞, such that τ₂−τ₁

2(t₂−t₁) = (3τ1)^1/3+ t₂−t₁

(3τ₁)^1/3 +2t₁t₂

3τ₁ +O 1 τ₁^5/3

;

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this specifies two new times t₁ and t₂. The following approximation, far out along the cusp, of the Pearcey process by the Airy process holds:

P

2

\

i=1

(P(τ_i)−₂₇²(3τ_i)^3/2

(3τi)^1/6 ∩(−E_i) = ∅ )!

= P

2

\

i=1

{A(t_i)∩(−E_i) =∅}

!

1 +O 1 τ₁^4/3

! .

Remark: The O τ₁^−4/3

-approximation, obtained via the PDE is much better than any rough estimate one might predict. Also one expects that, in this precise limit, the Pearcey process with inliers tends to the Airy process with outliers; see [2].

2 Non-intersecting Brownian motions and a chain of Coupled Random Matrices

Setting

τ_k :=t_k+1−t_k and 1

σ_k := 1

t_k−t_k−1 + 1

t_k+1−t_k, for 16k 6m, and taking in (1.1) the limitγ_i →0, for i= 1, . . . , N, leads to

P

m

\

k=1

n

all xi(tk)∈E˜k

o xj(0) = 0, xj(1) =δj, for j = 1, . . . , N

!

= 1

Z_n⁰ Z

˜E^N

∆_N(u₁)

m

Y

k=1

"

det

e

2uk;iuk+1;j τk

16i,j6N

Y

16i6N

e⁻

u2 k;i σk du_k;i

#

, (2.1) where ∆N(u1) stands for the Vandermonde determinant in the variablesu1 = (u_1;1, . . . , u_1;N). Notice that each of the sets of variables u₁, . . . , u_m appears in exactly two of the determinants in the above integrand and that the other factors are insensitive to a permutation, for fixed k with 1 6 k 6 m, of the variables u_k = u_k;1, . . . , u_k;N. Therefore, taking the limit u_m+1;i = δ_i → β_j, for i = 1, . . . , N, with n_` of the δ_i going to β_`, namely u_m+1;1, . . . , u_m+1;n₁ → β1, and so on, making m synchronized changes of variables, and using the

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symmetry of the integration ranges vis-`a-vis these variables u_k;1, . . . , u_k;N,

P







x_j(0) = 0, (j = 1, . . . , N), x₁(1) =· · ·=x_n₁(1) =β₁,

m

\

k=1

n

all xi(tk)∈E˜k

o ...

xN−np+1(1) =· · ·=x_N(1) =β_p







= 1

Z_n⁰⁰ Z

E˜^N

∆_N(u₁)

p

Y

`=1



∆_n_`(u^(`)_m)

n`

Y

i=1

e⁻

m

P

k=1 u(`)

k;i 2

σk +

m−1

P

k=1 2u(`)

k;iu(`) k+1;i τk +²^β`u

(`) m;i τm



 Y

16i6N 16k6m

du_k;i,

= 1

Z_n⁰⁰⁰ Z

E^N

∆_N(v₁)

p

Y

`=1

∆_n_`(v^(`)_m)

n`

Y

i=1

e⁻

m

P

k=1 1 2v_k;i^(`)²+

m−1

P

k=1

ckv^(`)_k;iv_k+1;i^(`) +b`v^(`)_m;i

! Y

16i6N 16k6m

dv_k;i,

=: 1 Z_n⁰⁰⁰

Z

E^N

I_n(v)

m

Y

k=1

dv_k, (2.2)

= 1

Z˜_n Z

spec(Mk)∈E_k

e⁻

1 2tr^„^P^m

k=1

M_k²−2

m−1

P

k=1

ckMkMk+1−2AMm

« m

Y

k=1

dM_k, (2.3)

where the diagonal matrix A, ck, ˜b` and ˜Ek were defined in (1.4) or alter- natively expressed below in terms of the σ_k’s and τ_k’s. The last integration is taken over Hermitian matrices, with spec(M_k) ∈ E_k. Also the change of integration variables u^(`)_k;i 7→v^(`)_k;i above is given by

v^(`)_k;i = r 2

σ_ku^(`)_k;i, c_k =

√σ_kσ_k+1 τ_k , b_` =

√2σ_m

τ_m β_`, E_k = r 2

σ_kE˜_k. For k = 1, . . . , m and for ` = 1, . . . , p, the vector u^(`)_k = (u^(`)_k;1, . . . , u^(`)_k;n

`) is defined by

(u_k;1, . . . , u_k;N) = (u⁽¹⁾_k;1, . . . , u⁽¹⁾_k;n

1, u⁽²⁾_k;1, . . . , u⁽²⁾_k;n

2, . . . , u^(p)_k;1, . . . , u^(p)_k;n

p).

Concerning the Jacobi matrix (1.8), one needs the following formulas for derivatives ofJ; they can be shown by recurrence:

c₁ ∂

∂c₁J_mm =−2J_m1² , c_m−1 ∂

∂cm−1

J_m1 =−J_m1(2J_mm+ 1). (2.4)

3 Integrable deformations

In this section, we introduce a time deformation ˜I_n(v) of the integrand I_n(v), introduced in (2.3). The deformation is chosen such that the resulting integral

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is on the one hand a solution to the multi-component KP hierarchy (see [8]

and Proposition 3.1 below) and satisfies on the other hand a set of Virasoro constraints. We will impose on the rescaled target pointsb₁, . . . , b_p, which we henceforth denote by b⁽¹⁾₁ , . . . , b^(p)₁ , a non-trivial linear constraint

p

X

`=1

κ_`b^(`)₁ = 0. (3.1)

Without loss of generality, we may assume (upon reordering) that κp 6= 0 and impose if Pp

1κ_` 6= 0 that Pp

`=1κ_` = 1; also define κ₀ := −1. Thus, the non-deformed integral which we will consider is

Z

E^N

I_n(v)|Pp

1κ`b^(`)_1,2=0 m

Y

k=1

dv_k. (3.2)

The integrand I_n(v) will be deformed by four sets of parameters: (i) A first set, denoted byb⁽¹⁾₂ , . . . , b^(p)₂ , deforms the parametersb^(`)₁ . They are subjected to the same constraint (3.1) as the parametersb^(`)₁ , namely¹¹

p

X

`=1

κ_`b^(`)₂ = 0. (3.3)

(ii) A second set of deformations consists of parameters corresponding to the KP time variables; they are denoted by s⁽⁰⁾r (r ∈ Z>0) for the parameters going with the starting point 0 of the Brownian motion and s^(`)r (1 6 ` 6 p andr ∈Z>0) for the parameters going with the`-th end point of the Brownian motion. (iii) There is furthermore a set of parameters γ^(k)r (2 6 k 6 m−1 and r∈Z^>0) going with the intermediate times t2, . . . , tm−1 and (iv) a set of parametersc^(k)r,q (k = 1, . . . , m−1 and¹²(r, q)>(1,1)), going with consecutive timest_k, t_k+1.

For n = (n₁, . . . , n_p) and E = E₁×E₂× · · · ×E_m, where each E_k is the union of a finite number of intervals in R, define

τ_n(E) :=

Z

E^N

I˜_n(v) Pp

1κ`b^(`)_1,2=0 m

Y

k=1

dv_k, (3.4)

11The combination of the two constraints (3.1) and (3.3) will in the formulas below be denoted byPp

1κ`b^(`)_1,2= 0.

12The inequality (r, q)>(1,1) means by definition thatr>1, q>1 and (r, q)6= (1,1).

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where

I˜_n(v) = I_n(v)×

p

Y

`=1 n`

Y

i=1

e

b^(`)₂ v^(`)_m;i²+P

r>1

(s⁽⁰⁾r v^(`)_1;i^r−s^(`)r v^(`)_m;i^r)+

m−1

P

k=1

P

(r,q)>(1,1)

c^(k)rqv^(`)_k;i^rv^(`)_k+1;i^q+

m−1

P

k=2

P

r>1

γr^(k)v_k;i^(`)^r

, with

I_n(v) = ∆_N(v₁) Qp

`=1n_`!

p

Y

`=1

∆_n_`(v_m^(`))

n`

Y

i=1

e

m−1

P

k=1

ckv^(`)_k;iv_k+1;i^(`) −¹

2 m

P

k=1

v^(`)_k;i²+b^(`)₁ v^(`)_m;i! .

We denote byLthe locus corresponding to setting all deformation parameters equal to zero, so that ˜In

_L=In,

L =











s⁽⁰⁾r , . . . , s^(p)r = 0, r∈Z>0, b⁽¹⁾₂ , . . . , b^(p)₂ = 0,

γr⁽²⁾, . . . , γr^(m−1) = 0, r ∈Z>0, c⁽¹⁾rq, . . . , c^(m−1)rq = 0, (r, q)>(1,1)











. (3.5)

We list a number of operator identities, valid when acting onτ_n(E),

∂

∂b^(`)_h = − ∂

∂s^(`)_h + κ_` κ_p

∂

∂s^(p)_h , 16` 6p−1, h= 1,2, (3.6)

p

X

`=1

b^(`)_j ∂

∂s^(`)_h = −

p−1

X

`=1

b^(`)_j ∂

∂b^(`)_h , h, j ∈ {1,2}, (3.7)

∂

∂s^(`)_h

= −(1−δ_`,p) ∂

∂b^(`)_h +κ_`

p

X

i=1

∂

∂s⁽ⁱ⁾_h +

p−1

X

i=1

∂

∂b⁽ⁱ⁾_h

!

= ∂_b^(`)

h +κ_`

p

X

i=1

∂

∂s⁽ⁱ⁾_h , h= 1,2, 16`6p, (3.8) where forh= 1,2 and 16`6p we define

∂_b^(`)

h :=−(1−δ_`,p) ∂

∂b^(`)_h +κ_`

p−1

X

i=1

∂

∂b⁽ⁱ⁾_h =

p−1

X

i=1

(κ_`−δ_`,i) ∂

∂b⁽ⁱ⁾_h , (3.9) implying

p

X

`=1

∂_b^(`)

h = 0. (3.10)

UsingPp

`=1κ_`b^(`)_h = 0, one first establishes identity (3.6) and then (3.7), while the first equality in (3.8) is obtained by computingPp−1

i=1

∂

∂b⁽ⁱ⁾_h from (3.6) and by using Pp

`=1κ_` = 1 and the identity (3.6).

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From section 7.3 in [8], it follows that τ_n(E) can be written as

τ_n(E) = det







(hyⁱϕ₁(y)|x^jψ(x)i) ⁰6i < n₁ 06j < N

...

(hyⁱϕp(y)|x^jψ(x)i) ⁰6i < n_p 06j < N





, (3.11)

where

ψ(x) := exp −1

2x²+X

r>1

s⁽⁰⁾_r x^r

! ,

ϕ_`(y) := exp −1

2y²+b^(`)₁ y+b^(`)₂ y²−X

r>1

s^(`)_r y^r

! ,

for` = 1, . . . , p, and where the inner product h· | ·i is defined by hf(y)|g(x)i:=

Z Z

E1×Em

f(y)g(x)µ(x, y)dx dy, with

µ(x, y) :=

Z

Qm−1 k=2 Ek

exp

m−1

X

k=2

−1

2w_k²+X

r>1

γ_r^(k)w^r_k

!

×

exp

m−1

X

k=1



c_kw_kw_k+1+ X

(r,q)>(1,1)

c^(k)_rqw^r_kw^q_k+1





m−1

Y

k=2

dw_k,

w₁ :=xandw_m :=y. Form = 2 the latter formula forµshould be interpreted as µ(x, y) := 1, while µ(x, y) := δ(x−y)e^x²^/2 (the delta distribution) in the case of m= 1.

The above representation (3.11) of τ_n implies, in view of [8, Prop. 6.2], that τn is a tau function of the p+ 1 component KP hierarchy, in particular we have the following Proposition.

Proposition 3.1 The functionτ_n =τ_n(E), as in (3.4), satisfies for16` 6p

∂

∂s⁽⁰⁾₁ lnτ_n+e_` τn−e_`

=

∂²

∂s⁽⁰⁾₂ ∂s^(`)₁ lnτ_n

∂²

∂s⁽⁰⁾₁ ∂s^(`)₁ lnτn

, ∂

∂s^(`)₁ lnτ_n+e_` τn−e_`

=−

∂²

∂s⁽⁰⁾₁ ∂s^(`)₂ lnτ_n

∂²

∂s⁽⁰⁾₁ ∂s^(`)₁ lnτn

, (3.12)

where n±e_` = (n₁, . . . , n_p)±e_` := (n₁, . . . , n`−1, n_`±1, n_`+1, . . . , n_p).

Both equations will play an important role in Section 6 below.

A PDE for non-intersecting Brownian motions and applications