On the Tracy-Widomβ Distribution for β=6
We study the Tracy-Widom distribution function for Dyson's β-ensemble with β=6. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions...
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| Cite this: | On the Tracy-Widomβ Distribution for β=6 / T. Grava, A. Its, A. Kapaev, F. Mezzadri // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. |
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| citation_txt | On the Tracy-Widomβ Distribution for β=6 / T. Grava, A. Its, A. Kapaev, F. Mezzadri // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. |
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| description | We study the Tracy-Widom distribution function for Dyson's β-ensemble with β=6. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of β. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom β=6 function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function. We also notice that our work is a partial answer to one of the problems related to the β-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 105, 26 pages
On the Tracy–Widomβ Distribution for β = 6?
Tamara GRAVA †1†2, Alexander ITS †
3
, Andrei KAPAEV †4 and Francesco MEZZADRI †
1
†1 School of Mathematics, University of Bristol, Bristol, BS8 1SN, UK
E-mail: tamara.grava@bristol.ac.uk, francesco.mezzadri@bristol.ac.uk
†2 SISSA, via Bonomea 265, 34100, Trieste, Italy
E-mail: grava@sissa.it
†3 Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis,
Indianapolis, IN 46202-3216, USA
E-mail: aits@iupui.edu
†4 Department of Mathematical Physics, St. Petersburg State University, St. Petersburg, Russia
E-mail: kapaev55@mail.ru
Received July 04, 2016, in final form October 25, 2016; Published online November 01, 2016
http://dx.doi.org/10.3842/SIGMA.2016.105
Abstract. We study the Tracy–Widom distribution function for Dyson’s β-ensemble with
β = 6. The starting point of our analysis is the recent work of I. Rumanov where he produces
a Lax-pair representation for the Bloemendal–Virág equation. The latter is a linear PDE
which describes the Tracy–Widom functions corresponding to general values of β. Using
his Lax pair, Rumanov derives an explicit formula for the Tracy–Widom β = 6 function in
terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov
also shows that this formula allows him to derive formally the asymptotic expansion of the
Tracy–Widom function. Our goal is to make Rumanov’s approach and hence the asymptotic
analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov’s
Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair
for the second Painlevé equation. This gauge transformation though contains functional
parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the
auxiliary ODE of Rumanov’s formula. The gauge-interpretation of Rumanov’s Lax-pair
allows us to highlight the steps of the original Rumanov’s method which needs rigorous
justifications in order to make the method complete. We provide a rigorous justification
of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov’s
formula is indeed, as it has been suggested by Rumanov, the Hastings–McLeod solution of
the second Painlevé equation. The key issue which we also discuss and which is still open is
the question of integrability of the auxiliary ODE in Rumanov’s formula. We note that this
question is crucial for the rigorous asymptotic analysis of the Tracy–Widom function. We
also notice that our work is a partial answer to one of the problems related to the β-ensembles
formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems.
Key words: β-ensamble; β-Tracy–Widom distribution; Painlevé II equation
2010 Mathematics Subject Classification: 30E20; 60B20; 34M50
Dedicated to Percy Deift and Craig Tracy on the occasion of their 70th birthdays
1 Introduction
Given β > 0, Dyson’s β-ensemble is defined as a Coulomb gas of N charged particles, that
is as the space of N one dimensional particles, {−∞ < λ1 < λ2 < · · · < λN < ∞} with the
?This paper is a contribution to the Special Issue on Asymptotics and Universality in Random Matrices,
Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy.
The full collection is available at http://www.emis.de/journals/SIGMA/Deift-Tracy.html
mailto:tamara.grava@bristol.ac.uk
mailto:francesco.mezzadri@bristol.ac.uk
mailto:grava@sissa.it
mailto:aits@iupui.edu
mailto:kapaev55@mail.ru
http://dx.doi.org/10.3842/SIGMA.2016.105
http://www.emis.de/journals/SIGMA/Deift-Tracy.html
2 T. Grava, A. Its, A. Kapaev and F. Mezzadri
probability density given by the equation
p(λ1, . . . , λN )dλ1 · · · dλN =
1
ZN
∏
1≤j,k≤N
|λj − λk|βe
−β
N∑
j=1
V (λ)
dλ1 · · · dλN , (1.1)
ZN =
∫ ∞
−∞
· · ·
∫ ∞
−∞
∏
1≤j,k≤N
|λj − λk|βe
−β
N∑
j=1
V (λ)
dλ1 · · · dλN .
Here, V (λ) has a meaning of external field which we will assume to be Gaussian, i.e., V (λ) = λ2
2 .
The objects of interest are the gap probabilities in the large N limit. We will be particularly
concerned with the soft edge probability distribution
Fβ(t) ≡ Esoft
β
(
0; (t,∞)
)
= lim
N→∞
Esoft
βN
(
0;
(√
2N +
t√
2N1/6
,∞
))
,
where
Esoft
βN
(
0; (t,∞)
)
=
∫ t
−∞
· · ·
∫ t
−∞
p(λ1, . . . , λN )dλ1 · · · dλN . (1.2)
Cases β = 1, 2, 4 known as Gaussian orthogonal (GOE), Gaussian unitary (GUE) and Gaussian
symplectic (GSE) ensembles. Indeed, in these cases, distribution (1.1) describes the statistics
of the eigenvalues of orthogonal, Hermitian, and symplectic random matrices, respectively, with
i.i.d. matrix entries. The corresponding limiting edge distribution functions Fβ(t) then become
the classical Tracy–Widom distributions [22]. They admit explicit representations as either
the Airy kernel Fredholm determinants or in terms of the Hastings–McLeod solution of the
second Painlevé equation. These representations, in turn, allow one to evaluate the asymptotic
expansions of Fβ(t) as t→ −∞.
In this paper we address the question of the asymptotic analysis of Fβ(t) beyond the classical
values β = 1, 2, 4. The crucial problem is that the orthogonal polynomial approach, which is
the principal technique in the intergrable random matrix case, is not available for general β.
However, several highly nontrivial conjectures concerning the general β ensembles have been
suggested. An excellent presentation of the state of art in this area is given in the survey by
P. Forrester [11]. The current principal heuristic result concerning the asymptotic behavior of
the generalized Tracy–Widom distribution Fβ(t) was obtained in 2010 by G. Borot, B. Eynard,
S.N. Majumdar and C. Nadal and it reads as follows.
Conjecture 1.1 ([3]).
Fβ(t) = exp
(
−β |t|
3
24
+
√
2
3
(β/2− 1)|t|3/2
+
1
8
(β/2 + 2/β − 3) log |t|+ c0 +O
(
1
|t|3/2
))
, t→ −∞. (1.3)
The constant term c0 is also explicitly predicted. Indeed, it is claimed that
c0 =
β
2
(
1
12
− ζ ′(−1)
)
+
γ
6β
− log 2π
4
− (β/2)
2
+
(
17
8
− 25
24
(β/2 + 2/β)
)
log 2 +
∫ ∞
0
1
eβt/2 − 1
(
t
et − 1
− 1 +
t
2
− t2
12
)
dt, (1.4)
where ζ(z) is Riemann’s zeta-function and γ denotes Euler’s constant.
On the Tracy–Widomβ Distribution for β = 6 3
Formulae (1.3), (1.4) have been derived in [3] within the framework of the so-called loop-
equation technique by performing the relevant double scaling limit directly in the formal large N
expansion of the multiple integral in (1.2).
Remark 1.2. For the classical cases β = 1, 2, 4, formula (1.3), without the constant term, was
obtained and proved in [22] and [23] by using the (established in these papers) representations
of Fβ=1,2,4(t) in terms of the Hastings–McLeod solution of the second Painlevé equation. More-
over, in [22, 23] the value (1.4) of the constant c0 for β = 1, 2, 4 was also conjectured. Rigorous
derivation of (1.3), (1.4) for the case β = 2 is given in [1, 5]. In [1], the cases β = 1, 4 are
also done. The papers [5] and [1] employ the Riemann–Hilbert approach and the Deift–Zhou
nonlinear steepest descent method [6] which are available to the classical cases of β = 1, 2, 4.
Remark 1.3. The leading asymptotic term in (1.3) for arbitrary β has been rigorously obtained
in [16] with the help of the analysis of the certain stochastic Schrödinger operator. We shall
mention this paper again right after the next remark.
Remark 1.4. For the limiting hard edge probability distribution, the heuristic asymptotic
result similar to (1.3) was obtained in [4]. In fact, in the case of positive integer values of
the exponent in the corresponding Lauguerre weight the asymptotic formula had been already
proven, including the rigorous derivation of the constant term, in [10]. For more on the hard
edge case we refer the interested reader again to survey [11].
It is remarkable, that paper [3], while giving such detailed formulae for the asymptotics of
the distribution function Fβ(t) does not actually produce any description of the object itself
(for finite values of t). The latter has been done by A. Bloemendal and B. Virág [2]. Inspired
by the pioneering work of E. Dumitriu and A. Edelman [7] and by the subsequent works [24]
and [16], Bloemendal and Virág [2] connect the analysis of the generalized Tracy–Widom distri-
bution Fβ(s) to the study of stochastic Schrödinger operators. In particular, it has been proven
in [2] that the Tracy–Widom distribution function Fβ(t), for any β, can be expressed in terms
of the solution of a certain linear PDE. In more details, the result of [2] can be formulated as
follows.
Consider the partial differential equation for the scalar function F (x, t;β)
∂F
∂t
+
2
β
∂2F
∂x2
+
(
t− x2
)∂F
∂x
= 0, (x, t) ∈ R2 (1.5)
supplemented by the boundary conditions
F (x, t;β)→ 1, as x, t→∞, (1.6)
and
F (x, t;β)→ 0, as x→ −∞ for fixed t. (1.7)
Theorem 1.5 ([2]). The boundary value problem (1.5)–(1.7) has a unique bounded smooth
solution. Moreover, equation
Fβ(t) = lim
x→∞
F (x, t;β),
determines the Tracy–Widom distribution function for the general value of the parameter β > 0.
Equation (1.5) has a very interesting interpretation from the point of view of the theory of
Painlevé transcendents. In fact, there are two ways to connect (1.5) with the Painlevé equations.
4 T. Grava, A. Its, A. Kapaev and F. Mezzadri
The first way identifies (1.5) with the quantum second Painlevé equation. Indeed, the second
Painevé equation, i.e., the equation
utt = tu+ 2u3, (1.8)
admits the Hamiltonian form [15]
ut =
∂H
∂p
, pt = −∂H
∂u
,
where the (time dependent) Hamiltonian H ≡ H(t, u, p) is
H =
p2
2
−
(
u2 +
t
2
)
p− u
2
.
Set
Φ(u, t;β) := F
(
−21/3e−i
π
3 u, 2−1/3ei
π
3 t;β
)
.
Then, in terms of Φ(u, t), equation (1.5) becomes the imaginary time Schrödinger equation
generated by the quantum Hamiltonian H(t, u, ~∂u) +u/2 with 2/β playing the role of Planck’s
constant ~. That is, we have that
2
β
∂tΦ =
(
H
(
t, u,
2
β
∂u
)
+
u
2
)
Φ.
Remark 1.6. Equation (1.5) in its original variables can be also interpreted as an imaginary
time Schrödinger equation generated by the quantum Hamiltonian H0(t, x, ~∂x) with
H0(t, x, p) := −p2 −
(
t− x2
)
p
and with the Planck constant ~ again equals 2/β. The corresponding classical dynamical system
is equivalent to the Painlevé 34 equation for the momentum p(t)
ptt = 4p2 + 2pt+
p2t
2p
,
which, in turn, is reduced again to the second Painlevé equation (1.8) for the function u =
√
p.
We refer the reader to the papers [14, 17, 20, 21, 26] for more on the quantum Painlevé
equations and their connections to the β-ensembles. It should be also mentioned that in the
current state of development of this subject, this connection provides us with no efficient tools
of asymptotic analysis of the β-ensembles.
The second connection of equation (1.5) to the Painlevé theory has been already found in [2],
and it gives indeed an efficient apparatus for the analysis of (1.5), but, at the moment, only
for the classical value β = 2. As it is shown in [2], if β = 2, the solution of (1.5) is reduced
to the solution of the Riemann–Hilbert problem canonically associated with the Painlevé II
equation (1.8). More precisely, this important observation can be described as follows.
The second Painlevé equation (1.8) is the compatibility condition of the following Flaschka–
Newell [8] Lax pair
dΨ0
dx
= L̂0Ψ0,
dΨ0
dt
= B̂0Ψ0, (1.9)
On the Tracy–Widomβ Distribution for β = 6 5
where the matrices L̂0 and B̂0 are
L̂0 =
x2
2
σ3 + x
(
0 u
u 0
)
+
− t2 − u2 −ut
ut
t
2
+ u2
and
B̂0 = −x
2
σ3 −
(
0 u
u 0
)
,
where σ3 =
(
1 0
0 −1
)
. Put
F (x, t;β = 2) = (Ψ0(x, t))22e
x3
6
− 1
2
tx+
∫∞
t ω(τ)dτ , ω = u4 + tu2 − u2t . (1.10)
Then F (x, t;β = 2) will satisfy (1.5) with β = 2. It is also shown in [2] that the boundary con-
ditions (1.6), (1.7) select in formula (1.10) the Hastings–McLeod [12] solution of (1.8) uniquely
defined by the asymptotic condition
u(t) ' Ai(t), t→ +∞,
and having the following behavior on the other end of the real line
u(t) ∼
√
− t
2
, t→ −∞.
Here, Ai(x) denotes the usual Airy function. This fact yields immediately the Tracy–Widom
formula for F2(t),
F2(t) = e
∫∞
t ω(τ)dτ ,
and, with the reference to [1, 5], the proof of Conjecture 1.1 above for β = 2.
It is tempting to find the analog of the Lax pair representation (1.10) for the solution of (1.5)
for arbitrary β. For the case of even values of β, important progress toward this goal has been
achieved by I. Rumanov in [18, 19]. In particular, in [19] Rumanov has produced a formula very
similar to (1.10) for the first interesting case, β = 6. However, Rumanov’s higher beta analogue
of the Lax pair (1.9) involves functional parameters which are defined via auxiliary nonlinear
ODEs, and this makes Rumanov’s approach for β > 4 less efficient than in the classical cases
of β = 1, 2, 4. Moreover, several important steps in Rumanov’s approach are only justified on
a heuristic level. In this manuscript we are suggesting a new interpretation of the results of [19]
which allows us to fill some of the gaps of Rumanov’s scheme.
Our first result is to show that Rumanov’s Lax pair in the case β = 6 can be obtained by a
gauge transformation of the Painlevé II Lax pair. Let Ψ0(x, t) be the fundamental solution of
the Painlevé II Lax pair (1.9) and let us introduce the matrix function Ψ(x, t) defined as
Ψ(x, t) := e
x3
6
−xt
2 κ(t)
1 + q2(t)
2
x− α(t) −1
1− q22(t)
4
0
e−
iπ
2
σ3
(
1
u(t)
)σ3
2
Ψ0(x, t), (1.11)
where κ(t), q2(t) and α(t) are free functional parameters and u(t) is the solution of the PII
equation (1.8). Then our key observation is that the matrix function Ψ(x, t) satisfies a Lax
pair equivalent to Rumanov’s Lax pair. Furthermore, using some of the constructions of [19]
6 T. Grava, A. Its, A. Kapaev and F. Mezzadri
presented in Section 2 of our paper, we arrive in Section 4 at the following conclusion. We show
that the equation
F (x, t;β = 6) = Ψ11
(
31/3x, 32/3t
)
, (1.12)
where the function Ψ(x, t) is given by (1.11), determines a solution of the Bloemendal–Virág
equation (1.5) for β = 6 if in addition one has
κt
κ
= −1
3
ω − 2
3
α− ut
u
1− 2q2
6
, (1.13)
and demands that the functions q2(t) and α(t) satisfy the following nonlinear ODEs
q2t = q2
(
2
3
α+
ut
u
2− q2
3
)
+
ut
u
2− q2
3
, (1.14)
and
αt = α
(
2
3
α+
ut
u
2− q2
3
)
− t
6
(1 + q2)−
u2
3
(3 + q2). (1.15)
Equations (1.13)–(1.15) are equivalent to the above mentioned auxiliary nonlinear ODEs of
Rumanov. The next question is how to reflect in this construction the boundary conditions
(1.6), (1.7).
Conjecture 1.7 ([19]). Equation (1.12) determines the solution of the boundary value problem
(1.5)–(1.7) if the Painlevé function u(t) is the Hastings–McLeod solution of (1.8) and the pair
(q2(t), α(t)) is the solution of the system (1.14)–(1.15) satisfying the following initial conditions
at t = +∞
q2 = −1 + o(1), α = o(1), t→ +∞. (1.16)
In addition, the function κ and the branch u1/2 in (1.11) should be f ixed so that
κu
1
2 → 1, as t→ +∞.
This conjecture should be supplemented by yet another conjecture concerning the system
(1.14), (1.15).
Conjecture 1.8. The system (1.14), (1.15) has a unique smooth solution (q2, α) that satisfy
conditions (1.16) as t→ +∞.
Assuming that Conjectures 1.7 and 1.8 are true, the Tracy–Widom distribution function for
β = 6 admits the following representation in terms of the Hastings–McLeod Painlevé func-
tion u(t) and the auxiliary function q2(t)
F6
(
3−2/3t
)
=
(q2 − 1)
2q2
exp
(
1
3
∫ ∞
t
ω(s)ds− 2
3
∫ ∞
t
us(s)
u(s)
1 + q2(s)
q2(s)
ds
)
, (1.17)
with ω as in (1.10). Although not identical, this formula is equivalent to Rumanov’s for-
mula (1.13) in [19].
The necessity to analyze the additional non-trivial differential equations, i.e., equations
(1.14), (1.15), makes representation (1.17) not quite good for an effective analysis. However, we
believe that this is an important step toward the rigorous theory of the β-ensembles with the
general value of β. In fact, as it is shown in [19], one can derive from (1.14), (1.15), at least for-
mally, a power series expansion of q2(t) as t→ −∞ which, when substituted into (1.17), would
On the Tracy–Widomβ Distribution for β = 6 7
match, except for the constant term χ, the asymptotic formula for F6(t) from Conjecture 1.1.
We reproduce this result in Section 8.
This paper is the first in a series where we intend to transform Rumanov’s approach into
a rigorous scheme. The main goal of this paper is to prove Conjecture 1.7, assuming that
Conjecture 1.8 is true. Our proof of Conjecture 1.7 is based on the already described observation
that Rumanov’s Lax pair is gauge equivalent by the transformation (1.11) to the standard Lax
pair (1.9) for the second Painlevé equation. The auxiliary functions q2(t), α(t) and κ(t) appear
as functional parameters of this gauge transformation. We expect that this gauge equivalence
to the Painlevé II Lax pair takes place for all Lax pairs which are found by Rumanov for even β.
Remark 1.9. The functional parameters used in [19] are denoted as q2(t), q1(t), q0(t) and U(t).
The relations to the parameters q2(t), α(t), u(t) and κ(t) which we use here are given by the
equations
q2 = q2, q1 = 2α+
ut
u
(1 + q2), q0 = 2α
ut
u
+ t+ 2u2,
U(t) = 6
d
dt
(
log
κ√
1− q22
)
− t2
2
.
Remark 1.10. We expect that the auxiliary ODEs (1.14) and (1.15) can be put within the
context of integrable systems and this will complete the analysis of the Tracy–Widom distri-
bution function for β = 6. We shall discuss this and other open issues related to Rumanov’s
approach in the concluding section of this paper where we shall also clarify our use of the term
“integrability”.
2 Rumanov’s Lax pair
The original Rumanov’s Lax pair is the following linear system of two 2× 2 matrix differential
equations
dΨ
dx
= LΨ,
dΨ
dt
= BΨ, (2.1)
where
L =
1
2
x2 − t+ x2q2 − xq1 + q0 2(x3 − x2e1 + xe2 − e3)
(x+ e1)
1− q22
2
+ q1q2 x2 − t− x2q2 + xq1 − q0
, (2.2)
and
B =
−x2 (1 + q2) + a −x2 + xb+ c
q22 − 1
4
−x
2
(1− q2) + d
, (2.3)
where
a = d+ q2(b− e1) + q1. (2.4)
Let us also introduce the functional parameter
U = 3(a+ d)− t2
2
. (2.5)
8 T. Grava, A. Its, A. Kapaev and F. Mezzadri
We note that
TrB = −x+
t2
6
+
U
3
.
The compatibility of equations (2.1) implies
dB
dx
− dL
dt
= [L,B], (2.6)
which is equivalent to the following set of equations for the parameters e1, e2, e3 and q0, q1
and q2:
de1
dt
= (b− e1)(q2e1 − q1) + q2(c+ e2)− q0, (2.7)
de2
dt
= −2 + q2(be2 + e3 − e1e2) + q1e2 + q1c− q0b, (2.8)
de3
dt
= e3(q1 − q2e1 + q2b) + q0c− b, (2.9)
dq0
dt
= −q2 +
1
2
e3
(
q22 − 1
)
+ c
(
q1q2 +
1
2
e1
(
1− q22
))
, (2.10)
dq1
dt
= −q1q2b+
1
2
(
q22 − 1
)
(e2 + be1 + c), (2.11)
dq2
dt
=
(
q22 − 1
)(
e1 −
1
2
b
)
− q1q2. (2.12)
We observe that the equations (2.7)–(2.12) fix only six of the nine free parameters introduced
to define the matrices L and B in (2.2) and (2.3) respectively. The parameters that still need
to be fixed are b, c and d, or equivalently, b, c and U introduced in (2.5).
The system of equations (2.7)–(2.12) has a set of integrals of motions that was obtained
in [19]. In order to define these integrals let us introduce the auxiliary functions
r2 =
q22 − 1
4
(
e21 − e2
)
− 1
2
e1q1q2 +
1
2
q2q0 +
1
4
q21, (2.13)
r1 =
q22 − 1
4
(e3 − e2e1) +
1
2
e2q1q2 −
1
2
q1q0, (2.14)
and
r0 =
q20
4
+ e1e3
q22 − 1
4
− 1
2
e3q1q2. (2.15)
Then the quantities
I0 = 2r0 + U − e1q2 + 2q1, I1 = 2r1 − 1− q2,
and
I2 = 2r2 + t
are the integrals of the system (2.7)–(2.12) [19]. A key observation now is the following statement.
Proposition 2.1 ([19]). The Lax pair (2.1) implies the differential identity
3(Ψ11)t + (Ψ11)xx +
(
t− x2
)
(Ψ11)x
= x2
(
r2 +
t
2
)
Ψ11 + x(3b− 2e1)Ψ21 + x
(
r1 −
1
2
− q2
2
)
Ψ11
+ (e2 + 3c)Ψ21 +
(
U
2
+
3
2
q2(b− e1) + q1 + r0
)
Ψ11.
On the Tracy–Widomβ Distribution for β = 6 9
An immediate consequence of this proposition is that the function
F (x, t;β = 6) = Ψ11
(
32/3t, 31/3x
)
satisfy the Bloemendal–Virág equation (1.5) for β = 6 if the following constraints are imposed
on the functional parameters of the Lax pair (2.1)
r2 = − t
2
, (2.16)
r1 =
1
2
+
q2
2
, (2.17)
b =
2
3
e1, (2.18)
c = −1
3
e2, (2.19)
U
2
+
3
2
q2(b− e1) + q1 + r0 = 0. (2.20)
Constraints (2.16) and (2.17) are the restrictions on the integrals I1 and I2:
I1 = I2 = 0. (2.21)
Constraint (2.20) can be also easily achieved; indeed, this is just a formula for the yet free
functional parameter U . Equations (2.18) and (2.19) are the genuine extra conditions on the
functional parameters of the Lax pair (2.1) which do not follow from the zero curvature equa-
tion (2.6). They make the relation of Rumanov’s Lax pair (2.1) to the Bloemendal–Virág
equation (1.5) with β = 6 not as straightforward as the relation of the standard Painlevé II Lax
pair to the Bloemendal–Virág equation (1.5) with β = 2. We will come back to this issue in
Section 4.
We conclude this section by noticing that together with (2.18), equation (2.20) implies that
the third remaining Rumanov’s integral is also zero,
I0 = 0.
3 WKB analysis of Rumanov’s Lax pair
In this section we present the large x asymptotic analysis of the solution Ψ(x, t) of the Lax
pair (2.1). Our consideration will be formal. The goal of this section is twofold. First, we want
to explain the WKB-meaning of Rumanov’s integrals I1 and I2. Secondly, the formulae obtained
here will serve as a motivation for the principal constructions of Section 4 which in turn will
allow us in Section 7 to obtain our main result – the proof of Conjecture 1.7.
The formal large x asymptotics of the function Ψ(x) is given by the following classical [25]
WKB-ansatz
ΨWKB(x) = T (x) exp
[∫
Λdx−
∫
diag
(
T−1
dT
dx
)
dx
]
, (3.1)
where the diagonal matrix Λ and the invertible matrix T are taken from the spectral decompo-
sition of the matrix L,
L = TΛT−1, Λ =
(
λ+ 0
0 λ−
)
.
10 T. Grava, A. Its, A. Kapaev and F. Mezzadri
The eigenvalues λ± are the roots of the characteristic equation
det(λ− L) ≡
(
λ− x2 − t
2
)2
−
(
x4
4
+ r2x
2 + r1x+ r0
)
= 0,
where r2, r1 and r0 are exactly the same as in (2.13)–(2.15). This means that the eigenvalues
are
λ± =
x2 − t
2
± µ, µ =
√
x4
4
+ r2x2 + r1x+ r0 =
x2
2
+ r2 +
r1
x
+ +
r0 − r22
x2
+ · · · ,(3.2)
and we also have
T =
(
Q(x) + µ Q(x)− µ
P (x) P (x)
)
=
1
2
x2(1 + q2)− q1x+ q0 + 2r2 + · · · (q2 − 1)x2 − q1x+ q0 − 2r2 + · · ·
−xq
2
2 − 1
2
+ q2q1 − e1
q22 − 1
2
−xq
2
2 − 1
2
+ q2q1 − e1
q22 − 1
2
,
where
P (x) = −(q22 − 1)(x+ e1)− 2q2q1
4
, Q(x) =
q2x
2 − q1x+ q0
2
.
In particular, we have that
T (x) =
(
x2 0
0 x
)
1 + q2
2
q2 − 1
2
1− q22
4
1− q22
4
[I +
M̂
x
+ · · ·
]
, (3.3)
where
M̂ =
(
α̃ α̃
α α
)
, α̃ =
1− q2
2(1 + q2)
(−q1 + (1 + q2)e1), α =
1 + q2
2(1− q2)
(q1 + (1− q2)e1).
Plugging the estimates (3.2) and (3.3) into the right hand side of equation (3.1), we arrive at
the following expansion as x→∞
ΨWKB(x) = x
σ3
2
1 + q2
2
q2 − 1
2
1− q22
4
1− q22
4
[I +
M
x
+ · · ·
]
e
x3
6
− tx
2 e
(
x3
6
+r2x
)
σ3+ν log xσ3 , (3.4)
where
ν = r1 −
q2
2
, M =
r22 − r0 − q1
2
α̃
α r0 − r22 +
q1
2
.
Note that
r2 = − t
2
+
1
2
I2 and ν =
1
2
+
1
2
I1.
This shows the role of the integrals I2 and I1: Integral I2 determines the exponential function
describing the essential singularity of the function Ψ(x) at x = ∞, while the integral I1 deter-
mines the formal exponent ν at x =∞. Also, taking into account conditions (2.21) we conclude
that
r2 = − t
2
and ν =
1
2
,
On the Tracy–Widomβ Distribution for β = 6 11
and rewrite expansion (3.4) as
ΨWKB(x) = x
σ3
2
1 + q2
2
q2 − 1
2
1− q22
4
1− q22
4
[I +
M
x
+ · · ·
]
e
x3
6
− tx
2 e
(
x3
6
− tx
2
)
σ3+
1
2
log xσ3 . (3.5)
Observe finally that expansion (3.5) with the proper modification of the matrix coefficient M ,
can be written in the form
ΨWKB(x) =
1
2
(1 + q2)x− α −1
1
4
(1− q22) 0
[I +
M0
x
+ · · ·
]
e
x3
6
− tx
2 e
(
x3
6
− tx
2
)
σ3 , (3.6)
where
M0 =
r22 − r0 −
q1
2
+ α 1
α
(
r0 − r22 +
q1
2
− α
)
r0 − r22 +
q1
2
− α
.
The formal series[
I +
M0
x
+ · · ·
]
e
(
x3
6
− tx
2
)
σ3 ,
characterizes the essential singularity of the canonical solutions of the auxiliary linear system
corresponding to the second Painlevé equation (see, e.g., [9]). Therefore, formula (3.6) suggests
that Rumanov’s Lax pair should be gauge equivalent to the standard Painlevé II Lax pair. The
exact description of this gauge equivalence will be given in the next section.
4 The gauge transformation to the Painlevé II Lax pair
Let Ψ0(x, t) be a (fundamental) solution of the Flaschka–Newell Painlevé II Lax pair, i.e.,
dΨ0
dx
= L̂0Ψ0,
dΨ0
dt
= B̂0Ψ0, (4.1)
where the matrices L̂0 and B̂0 are
L̂0 =
x2
2
σ3 + x
(
0 u
u 0
)
+
(
δ w
−w −δ
)
and
B̂0 = −x
2
σ3 −
(
0 u
u 0
)
.
The parameters u, w, δ are related by the equations
w = −ut, δ = − t
2
− u2.
The compatibility condition of the pair (4.1) is the second Painlevé equation (cf. (1.8))
utt = tu+ 2u3. (4.2)
12 T. Grava, A. Its, A. Kapaev and F. Mezzadri
We shall discuss the particular choice of the Painlevé II function u(t) and of the Ψ0 – function
later in Section 7.
Taking a hint from (3.6), we put
Ψ(x, t) := e
x3
6
−xt
2 κ(t)R(x, t)ψσ3(t)Ψ0(x, t),
where
R(x, t) =
(
p(t)x− α(t) −1
q(t) 0
)
.
Here, at the moment, κ, ψ, p, q and α are free functional parameters. The proof of the following
statement is straightforward though a bit tedious.
Proposition 4.1. The function Ψ(x, t) satisfies the Lax pair
dΨ
dx
= LΨ,
dΨ
dt
= BΨ,
where the matrices L and B are
L = Rψσ3L̂0ψ
−σ3R−1 +RxR
−1 +
(
x2
2
− t
2
)
I ≡ x3J + x2L2 + xL1 + L0, (4.3)
and
B = Rψσ3B̂0ψ
−σ3R−1 +RtR
−1 − x
2
I +
κt
κ
I +
ψt
ψ
Rσ3R
−1 ≡ −x2J + xB1 +B0, (4.4)
and the matrix coefficients Lk and Bk are given in terms of the Painlevé function u(t) and (still
free) functional parameters κ, ψ, p, q and α by the following equations:
J =
(
0
p
q
(1 + puψ2)
0 0
)
, (4.5)
L2 =
−puψ2 −α
q
− 2αp
q
uψ2 +
p2
q
wψ2
0 1 + puψ2
, (4.6)
L1 =
(uα− pw)ψ2 α2
q
uψ2 − 1
q
vψ−2 +
2p
q
δ − 2αp
q
wψ2
−quψ2 −(uα− pw)ψ2
, (4.7)
L0 =
wαψ2 − δ − t
2
−2αδ
q
+
α2
q
wψ2 − 1
q
yψ−2 +
p
q
−qwψ2 −wαψ2 + δ − t
2
, (4.8)
and
B1 =
puψ2 α
q
+
2αp
q
uψ2 +
pt
q
+ 2
ψt
ψ
p
q
0 −1− puψ2
, (4.9)
B0 =
κt
κ
− ψt
ψ
− αuψ2 −α
2
q
uψ2 +
1
q
vψ−2 − αt
q
− 2
ψt
ψ
α
q
quψ2 κt
κ
+
ψt
ψ
+ αuψ2 +
qt
q
. (4.10)
On the Tracy–Widomβ Distribution for β = 6 13
We want now to match the formulae (4.3)–(4.10) with the formulae (2.2), (2.3). From equa-
tion (4.5) we arrive at the first restriction
p
q
(
1 + puψ2
)
= 1. (4.11)
Also, taking again the hint from (3.6) we choose p and q in the form
p =
1 + q2
2
, q =
1− q22
4
,
where q2 is a new free functional parameter. With this choice, equation (4.11) transforms into
the relation
1 + q2
2
uψ2 = −1 + q2
2
,
and hence the formula for ψ:
ψ2 = −1
u
.
Therefore, the number of free parameters is reduced from five to three: q2, α, κ, and the final
formulae for the matrix coefficients Lk and Bk are the following:
J =
(
0 1
0 0
)
,
L2 =
1 + q2
2
4αq2
1− q22
+
ut
u
1 + q2
1− q2
0
1− q2
2
,
L1 =
−α−
ut
u
1 + q2
2
4
1− q22
(
−α2 + u2 + (1 + q2)δ − (1 + q2)α
ut
u
)
1− q22
4
α+
ut
u
1 + q2
2
,
L0 =
α
ut
u
− δ − t
2
4
1− q22
(
−2αδ + α2ut
u
+ utu+
1 + q2
2
)
−1− q22
4
ut
u
−αut
u
+ δ − t
2
,
and
B1 =
−
1 + q2
2
4
1− q22
(
−αq2 +
q2t
2
− ut
u
1 + q2
2
)
0 −1− q2
2
,
B0 =
κt
κ
+
1
2
ut
u
+ α
4
1− q22
(
α2 − u2 − αt + α
ut
u
)
−1− q22
4
κt
κ
− 1
2
ut
u
− α− 2q2q2t
1− q22
.
Comparing these formulae with Rumanov’s Lax pair (2.2), (2.3) we see that our L and B have
exactly the same structure with Rumanov’s parameters, q1, q0, e1, e2, e3, b, c, a, and d given in
terms of our parameters α, q2, and κ by the equations
q1 = 2α+
ut
u
(1 + q2), (4.12)
14 T. Grava, A. Its, A. Kapaev and F. Mezzadri
q0 = 2α
ut
u
− 2δ, (4.13)
e1 = − 4αq2
1− q22
− ut
u
1 + q2
1− q2
, (4.14)
e2 =
4
1− q22
(
−α2 + u2 + (1 + q2)δ − (1 + q2)α
ut
u
)
, (4.15)
e3 = − 4
1− q22
(
−2αδ + α2ut
u
+ utu+
1 + q2
2
)
, (4.16)
a =
κt
κ
+
1
2
ut
u
+ α, (4.17)
d =
κt
κ
− 1
2
ut
u
− α− 2q2q2t
1− q22
, (4.18)
b =
4
1− q22
(
−αq2 +
q2t
2
− ut
u
1 + q2
2
)
, (4.19)
c =
4
1− q22
(
α2 − u2 − αt + α
ut
u
)
. (4.20)
It is worth noticing that the relation
(L0)21 ≡ −
1− q22
4
ut
u
= e1
1− q22
4
+
q1q2
2
,
which is present in (2.2) follows from (4.12), (4.14) automatically. Also, automatically, we have
that
α =
1 + q2
2(1− q2)
(q1 + e1(1− q2)),
and relation (2.4). Moreover, the following statement is true (the proof is again straightforward).
Proposition 4.2. Let r2 and r1 are defined according to (2.13) and (2.14) where all the pa-
rameters e1, e2, e3, q1, and q0 are given as functions of q2, α, and u according to (4.12)–(4.16).
Then,
r2 = − t
2
and r1 =
q2
2
+
1
2
∀ q2, α, u.
From this Proposition it follows that the constraints (2.16) and (2.17) are satisfied automat-
ically while the equations (2.18), (2.19), and (2.20) must be imposed if we want the function
F (x, t;β = 6) := Ψ11
(
31/3x, 32/3t
)
to satisfy the Bloemendal–Virág equation. Equations (2.18) and (2.19) yield the ODEs for q2
and α while equation (2.20) produces formula for κ. In the next sections we will analyze (2.18),
(2.19), and (2.20).
5 ODEs for q2 and α
Consider first equation (2.18), i.e.,
b =
2
3
e1.
On the Tracy–Widomβ Distribution for β = 6 15
Substituting here (4.19) and (4.14) we arrive at the following differential equation for q2(t)
q2t = q2
(
2
3
α+
ut
u
2− q2
3
)
+
ut
u
2− q2
3
. (5.1)
Next, we look at equation (2.19), i.e.,
c = −1
3
e2.
Substituting here (4.20) and (4.15) we arrive at the following differential equation for α(t):
αt = α
(
2
3
α+
ut
u
2− q2
3
)
− t
6
(1 + q2)−
u2
3
(3 + q2). (5.2)
Note that (5.1) can be rewritten as
q2t =
2
3
αq2 +
ut
u
(1 + q2)(2− q2)
3
, (5.3)
which, in particular, yields the following formula for α in terms of q2
α =
3
2
q2t
q2
− ut
u
(1 + q2)(2− q2)
2q2
, (5.4)
and, in turn, allows us to transform the system (5.1), (5.2) of two first order ODEs to a single
second order ODE for the function q2:
q2tt =
2q2t
q2
(
q2t −
ut
u
)
+
4
9
(
q22 −
3
2
)(
u2t
u2
− t− 2u2
)
− 2
9
q2
(
3
u2t
u2
− t
)
+
4
9q2
u2t
u2
. (5.5)
Put
η =
2α
q2 − 1
− u2
ω
− ut
u
1 + q2
1− q2
, ω = u4 + tu2 − u2t .
Then equation (5.5) transforms to the following equation for the function η
9ηtt + 9ηηt + η3 + P (t)η +Q(t) = 0, (5.6)
where
P (t) = 12
(
u2
ω
− ω
)
t
− 4t, Q(t) =
2
3
Pt(t) +
2
3
.
This is equation (1.14) of Rumanov’s paper.
Using (4.12), we can pass from the pair (q2, α) to the pair (q2, q1). For the new unknowns,
equations (5.1), (5.2) transform to the following pair of equations
q2t =
1
3
q1q2 +
2
3
ut
u
(
1− q22
)
(5.7)
and
q1t = −2
3
ut
u
q1q2 +
1
3
q21 +
2
3
(
t− u2t
u2
)
+
2
3
(ut
u
)
t
q2. (5.8)
16 T. Grava, A. Its, A. Kapaev and F. Mezzadri
These are equations (3.17) and (3.18) of paper [19] (we note that our u(t) is Rumanov’s q(t)).
A very interesting fact shown in [19] is that these equations can be linearized. Consider the
following 3× 3 linear system for the new functions µ± and ν,
µ+t =
2
3
ut
u
µ+ −
1
3
ν, (5.9)
µ−t = −2
3
ut
u
µ− +
1
3
ν, (5.10)
νt =
2
3
u2µ− +
2
3
ω
u2
µ+ (5.11)
Then, as it is shown in Lemma 2 of [19], the formulae
q2 =
µ+ + µ−
µ+ − µ−
and q1 =
2ν
µ+ − µ−
(5.12)
determine solution of the nonlinear system (5.7), (5.8). It is worth reproducing the proof of
Rumanov’s lemma.
Put
µ :=
µ+ + µ−
2
and χ :=
µ+ − µ−
2
.
Then, the system (5.9)–(5.11) can be rewritten as
µt =
2
3
ut
u
χ, (5.13)
χt =
2
3
ut
u
µ− 1
3
ν, (5.14)
νt =
2
3
(ut
u
)
t
µ+
2
3
(
t− u2t
u2
)
χ, (5.15)
where we have taken into account the relation (consequence of the Painlevé equation)
u2 +
ω
u2
=
(ut
u
)
t
.
Simultaneously, formulae (5.12) become the formulae
q2 =
µ
χ
and q1 =
ν
χ
, (5.16)
assuming that µ, χ, and ν satisfy (5.13)–(5.15). Then, from (5.16) we would have that
q2t =
µt
χ
− µ
χ2
χt =
2
3
ut
u
− µ
χ
(
2
3
ut
u
µ
χ
− 1
3
ν
χ
)
=
2
3
ut
u
− q2
(
2
3
ut
u
q2 −
1
3
q1
)
,
which is identical to (5.7). Similarly, we have that
q1t =
νt
χ
− ν
χ2
χt =
2
3
(ut
u
)
t
ν
χ
+
2
3
(
t− u2t
u2
)
− ν
χ
(
2
3
ut
u
µ
χ
− 1
3
ν
χ
)
=
2
3
(ut
u
)
t
q1 +
2
3
(
t− u2t
u2
)
− q1
(
2
3
ut
u
q2 −
1
3
q1
)
,
which is (5.8).
The possibility to linearize the auxiliary nonlinear system (5.7), (5.8), and hence the auxiliary
Lax pair constraints (2.18) and (2.19), is, of course, a very important observation. However,
On the Tracy–Widomβ Distribution for β = 6 17
this, by itself, does not put the β = 6 case on the same footing as the case β = 2. Indeed,
the original Bloemendal–Virág equation (1.5) is already a linear differential equation. True, it
is a PDE equation and not a system of ODE equations. Still, to make Rumanov’s approach
a really efficient scheme, one needs to put the system (5.9)–(5.11) in an integrable context. In
fact, this could be crucial for proving Conjectures 1.8 and 1.1. We shall say more about this
issue in the concluding section of the paper.
One of the possible ways to “integrate” equations (5.9)–(5.11) would be to express the func-
tions µ±(t) and ν(t) in terms of the solution Ψ0(x, t) of the Painlevé II Lax pair (4.1). In fact, it
is not necessary to try to reduce (5.9)–(5.11) to (4.1). Any alternative Lax pairs for the second
Painlevé equation would do. We shall analyze the possibility to connect the system (5.9)–(5.11)
to one of the known Lax pairs for Painlevé II in our future publication.
Remark 5.1. As it was pointed out to us by A. Prokhorov, equation (5.6) can be linearized
even more directly. One can easy check that the substitution
η = 3
d ln f
dt
,
transforms (5.6) to the following third order linear equation for the function f(t)
27fttt + 3P (t)ft +Q(t)f = 0.
This fact by itself does not lead, however, to the proof of Conjecture 1.1 – see discussion in the
concluding section of the paper.
6 Formulae for r0, κ, and U(t). The third Rumanov’s integral
In this short section we present expressions for κ and U which follow from (2.20) and also
establish that, as expected, the third Rumanov’s integral, I0 vanishes.
Substituting into (2.15) equations (4.12), (4.13), (4.14), and (4.16), we arrive at the following
formula for r0
r0 = ω +
t2
4
− ut
u
1 + q2
2
, (6.1)
where
ω := u4 + tu2 − u2t .
Note that −1
2ω is the Hamiltonian of the second Painlevé equation (4.2), and also that
ωt = u2.
In its turn, equation (6.1) together with the equations (4.17) and (4.12), after having been
substituted into (2.20), yields the formula for κ
κt
κ
= −1
3
ω − 2
3
α− ut
u
1− 2q2
6
. (6.2)
At the same time, from (4.17), (4.18), (2.4) and (2.5), we have that
U(t) = 6
κt
κ
− 6
q2q2t
1− q22
− t2
2
,
18 T. Grava, A. Its, A. Kapaev and F. Mezzadri
or, taking into account (6.2),
U(t) = −2ω − 4α− ut
u
(1− 2q2)− 6
q2q2t
1− q22
− t2
2
.
Furthermore, taking into account ODE (5.3), we can transform the last equation into the equa-
tion
U(t) = −2ω − 4
1− q22
α− ut
u
1 + q2
1− q2
− t2
2
. (6.3)
Now, we can check the third Rumanov’s integral, i.e.,
I0 =
q20
2
+ e1e2
q22 − 1
2
− e3q1q2 + U − e1q2 + 2q1.
This must be zero. We notice that it is related to our function r0 by the equation
I0(t) = 2r0 + U − e1q2 + 2q1.
Substituting here (6.1), (6.3) and (4.12) we check that indeed
I0(t) ≡ 0.
7 β = 6 Tracy–Widom function. The proof of Conjecture 1.7
The results of the previous sections can be formulated as the following theorem
Theorem 7.1. Let u(t) and Ψ0(x, t) be the solutions of the second Painlevé equation (4.2), and
of the Lax pair (4.1), respectively. Let also the functions q2(t) and α(t) be the solutions of the
differential equations (5.1) and (5.2), respectively. Finally, define the function κ(t) according to
equation (6.2). Then equations
F
(
3−1/3x, 3−2/3t;β = 6
)
= (Ψ(x, t))11,
Ψ(x, t) = κe
x3
6
−xt
2 R(x, t)e−
iπ
2
σ3u−
1
2
σ3Ψ0(x, t), (7.1)
where
R(x, t) =
1 + q2
2
x− α −1
1− q22
4
0
define a solution F (x, t;β = 6) of the Bloemendal–Virág equation (1.5). These formulae can be
written as the following single equation
F
(
3−1/3x, 3−2/3t;β = 6
)
= −iκe
x3
6
−xt
2
[
u−
1
2
(
1 + q2
2
x− α
)
Ψ011(x, t) + u
1
2 Ψ021(x, t)
]
. (7.2)
Our goal now is to fix solutions of the Painlevé II equation and of the equations (5.1) and (5.2)
so that equation (7.2) defines the unique solution of (1.5) which produces the β = 6 Tracy–
Widom distribution function. We remind that this solution is uniquely determined by the
following boundary conditions
F (x, t;β)→ 1, as x, t→ +∞, (7.3)
F (x, t;β)→ 0, as x→ −∞, t ≤ t0 <∞. (7.4)
On the Tracy–Widomβ Distribution for β = 6 19
The β = 6 Tracy–Widom distribution function F6(t) is then given by the equation
F6(t) = lim
x→+∞
F (x, t;β = 6).
Let us translate the boundary conditions (7.3), (7.4) to the relevant boundary conditions for the
functions Ψ0, q2 and α. We start with Ψ0.
The linear system
dΨ0
dx
= L̂0Ψ0
has six canonical solutions, Ψ
(k)
0 (x) which are characterized by the following asymptotic behavior
in the complex x-plane (for all the details see, e.g., [9])
Ψ
(k)
0 (x) ∼
(
I +
m1
x
+ · · ·
)
e
(
x3
6
−xt
2
)
σ3 , x→∞, (7.5)
π
2
+
k − 2
3
π < arg x <
π
2
+
k
3
π, k = 1, 2, . . . , 7.
The second Painlevé function u(t) can be recovered from the coefficient m1 of the expansion (7.5)
u = −m1,12 = m1,21. (7.6)
The canonical solutions are entire functions of x and they are related by Stokes matrices
Ψ
(k+1)
0 (x) = Ψ
(k)
0 (x)S
(k)
0 , k = 1, 2, . . . , 6. (7.7)
Also, one has
Ψ
(7)
0 (x) = Ψ
(1)
0 (x). (7.8)
The Stokes matrices have the following structure:
S
(1)
0 =
(
1 0
−is1 1
)
, S
(2)
0 =
(
1 is2
0 1
)
, S
(3)
0 =
(
1 0
−is3 1
)
,
S
(4)
0 =
(
1 −is1
0 1
)
, S
(5)
0 =
(
1 0
is2 1
)
, S
(6)
0 =
(
1 −is3
0 1
)
. (7.9)
where s1, s2 , s3 can be any complex numbers subject the cyclic relation, which follows from (7.8)
s1 − s2 + s3 + s1s2s3 = 0.
We shall restrict ourselves by considering only the real Painlevé functions u, which is equivalent
to the additional restrictions on the Stokes parameters
s1 = s3, s2 = s2.
Moreover, we shall be concerned with the Ablowitz–Segur family of the solutions which means
the further restrictions
s2 = 0, s1 = −ia = −s3, a ∈ R, |a| ≤ 1.
For all a, the solution u(t) decays exponentially as t→ +∞. In fact one has
u(t) =
a
2
√
π
t−
1
4 e−
2
3
t
3
2 (1 + o(1)), t→ +∞. (7.10)
20 T. Grava, A. Its, A. Kapaev and F. Mezzadri
If |a| < 1, then the solution u(t) decays and oscillates as t → −∞, while if |a| = 1 (the
Hastings–McLeod solution), the function u(t) grows as |t|1/2
u(t) = ±
√
− t
2
+O
(
1
t
)
, t→ −∞.
We shall show now that in order to formulae (7.1) produce the solution of the Bloemendal–Virág
equation (1.5) satisfying the boundary conditions (7.3), (7.4) one has to choose the Hastings–
McLeod Stokes data and to take Ψ0 in (7.1) as
Ψ0(x, t) = iΨ
(6)
0 (x, t)σ1, σ1 =
(
0 1
1 0
)
. (7.11)
Set
Y (6)(x, t) := Ψ
(6)
0 (x, t)e
−
(
x3
6
−xt
2
)
σ3 . (7.12)
We would have that, uniformly for all t > 0
Y (6)(x, t) ∼ I +
m1(t)
x
+ · · · , x→ +∞ (7.13)
(cf. (7.5)). Substituting (7.11)–(7.13) into (7.2) we have that
F
(
3−1/3x, 3−2/3t;β = 6
)
= κe
x3
6
−xt
2
[
u−
1
2
(
1 + q2
2
x− α
)
Ψ
(6)
012(x, t) + u
1
2 Ψ
(6)
022(x, t)
]
= κu
1
2
[
u−1
(
1 + q2
2
x− α
)
Y
(6)
12 (x, t) + Y
(6)
22 (x, t)
]
. (7.14)
From (7.14), in view of (7.13) and (7.6), we then get that
F
(
3−1/3x, 3−2/3t;β = 6
)
= κu
1
2
(
1− q2
2
+O
(
1
x
))
, x→ +∞, (7.15)
uniformly for t > 0. Assume now, that equations (5.1), (5.2) admit the solutions with the
following behavior as t→ +∞
q2(t) = −1 + o(1), α = o(1), t→ +∞. (7.16)
Taking also into account the exponential decay (7.10) of the Hastings–McLeod solution u(t) of
the Painlevé equation (4.2), we conclude from (6.2) that
κt
κ
∼ −1
2
ut
u
, t→ +∞.
Therefore, κ(t) can be defined in such a way that
κu
1
2 → 1, t→ +∞.
This, together with estimate (7.15), implies the first boundary condition (7.3) for the function
F (x, t;β).
To see what we have for the second boundary condition we first use the Stokes equations (7.7)
and the triviality of the Stokes matrix S
(2)
0 to rewrite Ψ
(6)
0 (x, t) as
Ψ
(6)
0 (x, t) = Ψ
(3)
0 (x, t)S
(3)
0 S
(4)
0 = Ψ
(3)
0 (x, t)
(
1 0
a 1
)(
1 −a
0 1
)
= Ψ
(3)
0 (x, t)
(
1 −a
a 1− a2
)
. (7.17)
On the Tracy–Widomβ Distribution for β = 6 21
Write (cf. (7.12))
Y (3)(x, t) := Ψ
(3)
0 (x, t)e
−
(
x3
6
−xt
2
)
σ3 . (7.18)
We would have that, this time, for every finite t
Y (3)(x, t) ∼ I +
m1(t)
x
+ · · · , x→ −∞. (7.19)
Substituting (7.11), (7.17), (7.18), and (7.19) into (7.2) we will obtain an alternative to (7.14)
representation for the function F (x, t;β)
F
(
3−1/3x, 3−2/3t;β = 6
)
= (1− a2)κu
1
2
[
u−1
(
1 + q2
2
x− α
)
Y
(3)
12 (x, t) + Y
(3)
22 (x, t)
]
− aκu
1
2 e
x3
3
−xt
[
u−1
(
1 + q2
2
x− α
)
Y
(3)
11 (x, t) + Y
(3)
21 (x, t)
]
. (7.20)
Taking into account that
e
x3
3
−xt → 0, x→ −∞, t ≤ t0 <∞,
and that a2 = 1 for the Hastings–McLeod solution, we arrive at the second boundary condi-
tion (7.4) for the function F (x, t;β).
Our analysis can be summarized as the following proposition.
Proposition 7.2. Let u(t) be the Hastings–McLeod solution of the second Painlevé equa-
tion (4.2) and Ψ(6) be the canonical solution of the corresponding isomonodromy linear problem.
Suppose that equations (5.1), (5.2) have solutions which are smooth for all real t and satisfy
conditions (7.16) at t = +∞. Then formulae (7.2), (7.11) define the (unique) solution of the
Bloemendal–Virág equation (1.5) satisfying the boundary conditions (7.3), (7.4). This, in turn,
yields the following formula for the β = 6 Tracy–Widom distribution function
F6
(
3−2/3t
)
=
1− q2
2
exp
(
1
3
∫ ∞
t
ω(s)ds+
2
3
∫ ∞
t
α(s)ds− 1
3
∫ ∞
t
us(s)
u(s)
(1 + q2(s))ds
)
,
which, taking into account (5.4), can be also written as
F6
(
3−2/3t
)
=
(q2 − 1)
2q2
exp
(
1
3
∫ ∞
t
ω(s)ds− 2
3
∫ ∞
t
us(s)
u(s)
1 + q2(s)
q2(s)
ds
)
. (7.21)
Formula (7.21) is equivalent, though not identical, to the original formula (1.13) of [19].
8 Asymptotics of F6(t) as t→ −∞
In this section we show that, at least on the formal level, equation (7.21) can be used to evaluate
the asymptotics of F6(t) as t→ −∞. Similar fact involving the original Rumanov’s formula has
already been demonstrated in [19].
To make step towards formula (1.3) we have to find asymptotics of all the integrands in (7.21).
The first ingredient of our computation is the well-known formal asymptotic expansion of the
Hastings–McLeod solution
u(t) =
√
− t
2
(
1− 1
8
(−t)−3 − 73
128
(−t)−6 − 10657
1024
(−t)−9
− 13912277
32768
(−t)−12 − 8045883943
262144
(−t)−15 − 14518451390349
4194304
(−t)−18 + · · ·
)
,
22 T. Grava, A. Its, A. Kapaev and F. Mezzadri
and the expansion of its logarithmic derivative
ut(t)
u(t)
= − 1
2(−t)
− 3
8
(−t)−4 − 111
32
(−t)−7 − 1509
16
(−t)−10
− 2617599
512
(−t)−13 − 944695983
2048
(−t)−16 − 127756233309
2048
(−t)−19 + · · · . (8.1)
The above expansions immediately yield the formal expansion of the Hamiltonian function ω
defined in (6.1)
ω(t) = −1
4
(−t)2 − 1
8
(−t)−1 − 9
64
(−t)−4 − 189
128
(−t)−7 − 21663
512
(−t)−10
− 4825971
2048
(−t)−13 − 3540311739
16384
(−t)−16 − 241980297111
8192
(−t)−19 + · · · . (8.2)
In contrast, the formal expansion of q2(t) as t→ −∞ is much less straightforward and requires
relatively significant efforts. With this aim, we utilize the Rumanov’s linearization (5.9)–(5.11)
and then apply (5.12).
The corresponding coefficient matrixM(t) of the vector equation for ~µ := (µ+, µ−, ν)T , i.e.,
~µt =M~µ,
M(t) =
2
3
ut
u
0 −1
3
0 −2
3
ut
u
1
3
2
3
ω
u2
2
3
u2 0
=
− 1
3(−t)
+O(t−4) 0 −1
3
0
1
3(−t)
+O(t−4)
1
3
−(−t)
3
+O
(
t−2
) (−t)
3
+O
(
t−2
)
0
degenerates in the leading order at infinity. Thus we first apply the shearing gauge transforma-
tion
~µ = R0~y, R0 =
(−t)−1/4 0 0
0 (−t)−1/4 0
0 0 (−t)1/4
1 1 1
1 −1 −1
0
√
2 −
√
2
,
that diagonalizes in the leading order the coefficient matrix
~yt = N~y, N = R−1MR−R−1Rt
=
−
1
4(−t) +O
(
t−20
)
− 1
3(−t) +O
(
t−7
)
− 1
3(−t) +O
(
t−4
)
− 1
6(−t) +O
(
t−5/2
)
−
√
2
3
√
−t+O
(
t−5/2
)
− 1
4(−t) +O
(
t−5/2
)
− 1
6(−t) +O
(
t−5/2
)
− 1
4(−t) +O
(
t−5/2
) √
2
3
√
−t+O
(
t−5/2
)
,
that enables us to construct effectively the formal asymptotic expansion of ~y(t) as t → −∞.
Three independent vector solutions to the above linear ODE form a matrix Y (t)
Y (t) =
(
I +
∞∑
k=1
Yk(−t)−3k/2
)(−t)1/4 0 0
0 e
2
√
2
9
(−t)3/2 0
0 0 e−
2
√
2
9
(−t)3/2
,
On the Tracy–Widomβ Distribution for β = 6 23
where Yj are independent from t matrix coefficients
Y1 =
0
1√
2
− 1√
2
− 1
2
√
2
− 11
48
√
2
− 3
8
√
2
1
2
√
2
3
8
√
2
11
48
√
2
, Y2 =
−1
9
259
96
259
96
1
2
− 665
3072
113
256
1
2
113
256
− 665
3072
,
Y3 =
0
83803
3072
√
2
− 83803
3072
√
2
− 347
72
√
2
− 1733015
1327104
√
2
− 60101
24576
√
2
347
72
√
2
60101
24576
√
2
1733015
1327104
√
2
, . . . .
The gauge transformation R0 does not mix the vector columns, thus we have three possible
solutions q2(t) distinguished by their asymptotic behavior as t→ −∞ according to which basic
vector solution ~y dominates in the relevant combination
q2(t) =
(−t)3 − 225
4
+O
(
t−3
)
, ~y ∼ (−t)1/4,
1√
2
(−t)−3/2 +
21
8
(−t)−3 +
1707
64
√
2
(−t)−9/2 +O
(
t−6
)
, ~y ∼ e
2
√
2
9
(−t)3/2 ,
− 1√
2
(−t)−3/2 +
21
8
(−t)−3 − 1707
64
√
2
(−t)−9/2 +O
(
t−6
)
, ~y ∼ e−
2
√
2
9
(−t)3/2 .
(8.3)
Finally, using (8.1), (8.2) and the second choice in the expansion (8.3) (this means that in an
exact description of q2(t), the dominant vector ~µ is presented) we find that
d
dt
logF6
(
3−2/3t
)
=
1
12
t2 −
√
2
3
(−t)1/2 +
1
24 t
+O
(
|t|−
5
2
)
as t→ −∞,
so that by the scaling t→ 32/3t one obtains
d
dt
logF6(t) =
3
4
t2 −
√
2(−t)1/2 +
1
24t
+O
(
|t|−
5
2
)
as t→ −∞,
which after integration gives
logF6(t) = −1
4
|t|3 +
2
√
2
3
|t|3/2 +
1
24
log |t|+ c0 +O
(
|t|−
3
2
)
as t→ −∞, (8.4)
for some constant c0. The above expression coincides with (1.3) for β = 6. In our derivation the
quantity χ is an undetermined constant.
9 Open questions
In this final section we highlight the two principal open questions in our version of Rumanov’s
scheme which are needed to be answered in order to make the method complete.
1. Prove that indeed the system (5.1), (5.2) has global smooth solution satisfying Cauchy
conditions (7.16) at t = +∞.
24 T. Grava, A. Its, A. Kapaev and F. Mezzadri
2. Assuming that the previous problem has been solved, establish that the solution q2(t) has
the power expansion as t → −∞ which generates via equation (7.21) the asymptotics for
Fβ=6(t) obtained in [3]. Here, the main challenge is to prove that the solution with the
Cauchy data q2(+∞) = −1 and q2t(+∞) = 0 has indeed the needed power series expansion
at t = −∞. This is a connection problem, and we strongly believe that it would be very
difficult to solve it without establishing the Lax-integrability of equations (5.1), (5.2). It
is important to emphasize that formally, the needed expansions of q2(t) at t = +∞ and
t = −∞ could be found by a direct perturbation analysis of equations (5.1), (5.2). For
that, integrability is really not needed; indeed, this has already been done in [19] and
demonstrated in Section 8 of this paper. The real issue is to prove that these expansions
are expansions of the same solution.
Both problems indicated above, will be solved if, for instance, one succeeds in the reduction
of the linear version of equations (5.1), (5.2), i.e., of the equations (5.9)–(5.11) to one of the
known Lax pairs for the second Painlevé equation as it is discussed at the end of Section 5. We
intend to address all these questions in our next publication as well as the issue of the extension
of these results to the all even values of β.
Before concluding this paper we want to make some extra comments on the linearizability of
equations (5.1), (5.2) and on the relevance of this fact to our principal goal, i.e., to the proof
of Conjecture 1.1. The fact that these equations, as well as equation (5.5) and Rumanov’s
equation (5.6), are linearizable, is, of course, very important, but in itself is not enough to
solve the above mentioned connection problem and hence to prove Conjecture 1.1. Indeed,
usually, in order to solve connection problems1 for a linear equation with rational coefficients,
one needs to have some additional information about its solutions. Most often this addition
information is given in the form of contour integral representation which is available through
the Laplace’s method and only for very special linear equations, i.e., for hypergeometric equation
and its degenerations. In the nonlinear case, or in the case of linear equations with meromorphic
coefficients (as it is the case with equations (5.9)–(5.11)), Laplace’s method is replaced by the
Riemann–Hilbert method and the contour integral representation is replaced by the Riemann–
Hilbert representation. The Riemann–Hilbert method is as effective for solving connection
problems for nonlinear equations as Laplace’s method in the linear case (see, e.g., [9]). However,
for the applicability of the Riemann–Hilbert method one needs Lax pairs. Hence our desire to
have a Lax-pair formulation either for equations (5.1), (5.2) themselves or for their linear version
(5.9)–(5.11).
We also want to mention one more interesting observation. Linearizability of the second order
differential equations (5.5) and (5.6) mean that they possess the Painlevé property2 and hence
must be equivalent to one of the 50 canonical equations from the Gambier list, see [13]. Let us
take Rumanov’s equation (5.6) and make the following substitutions,
η(t) = λ(t)W (z) + ζ(t), z = ξ(t),
where the local analytic change-of-variable functions λ(t), ζ(t), and ξ(t) are defined through the
equations,
9ζtt + 9ζζt + ζ3 + P (t)ζ +Q(t) = 0, (9.1)
9λtt + 9ζλt +
(
9ζt + 3ζ3 + P (t)
)
λ = 0,
1By solving a connection problem we mean to solve it explicitly, that is in terms of elementary or known
special functions, i.e., exactly in the form which we need solution of our problem in order to prove (8.4) and
Conjecture 1.1.
2The solutions of these equations do not have movable branch points; all their possible branch points are at
the poles of the coefficients of the equations. Indeed, in [19] all the relevant exponents have been calculated and
the absence of logarithmic terms has been established.
On the Tracy–Widomβ Distribution for β = 6 25
and
ξt =
1
3
λ.
Then, equation (5.6) transforms to the equation
d2W
dz2
= −3W
dW
dz
−W 3 + v(z)
{
dW
dz
+W 2
}
, (9.2)
where
v = −3ζλ+ 9λt
λ2
.
This is equation # VI from the list given in the Ince monograph [13]. This equation is linearized
by the substitution, W = −d lnF
dz . Moreover, the equation on F is
Fzzz = vFzz,
and hence is solvable in quadratures. Unfortunately, one of the change – of variables equations –
equation (9.1), is again Rumanov’s equation (5.6). Hence, though theoretically important, the
reduction of (5.6) to (9.2) does not immediately help in the achievement of our main goal, i.e.,
to prove (8.4).
Acknowledgements
A. Its and T. Grava acknowledge the support of the Leverhulme Trust visiting Professorship
grant VP2-2014-034. A. Its acknowledges the support by the NSF grant DMS-1361856 and by
the SPbGU grant N 11.38.215.2014. A. Kapaev acknowledges the support by the SPbGU grant
N 11.38.215.2014. F. Mezzadri was partially supported by the EPSRC grant no. EP/L010305/1.
T. Grava acknowledges the support by the Leverhulme Trust Research Fellowship RF-2015-442
from UK and PRIN Grant “Geometric and analytic theory of Hamiltonian systems in finite and
infinite dimensions” of Italian Ministry of Universities and Researches.
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1 Introduction
2 Rumanov's Lax pair
3 WKB analysis of Rumanov's Lax pair
4 The gauge transformation to the Painlevé II Lax pair
5 ODEs for q2 and
6 Formulae for r0, , and U(t). The third Rumanov's integral
7 =6 Tracy–Widom function. The proof of Conjecture 1.7
8 Asymptotics of F6(t) as t-
9 Open questions
References
|
| id | nasplib_isofts_kiev_ua-123456789-148536 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:56:31Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Grava, T. Its, A. Kapaev, A. Mezzadri, F. 2019-02-18T14:51:20Z 2019-02-18T14:51:20Z 2016 On the Tracy-Widomβ Distribution for β=6 / T. Grava, A. Its, A. Kapaev, F. Mezzadri // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30E20; 60B20; 34M50 DOI:10.3842/SIGMA.2016.105 https://nasplib.isofts.kiev.ua/handle/123456789/148536 We study the Tracy-Widom distribution function for Dyson's β-ensemble with β=6. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Virág equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of β. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom β=6 function in terms of the second Painlevé transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlevé equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlevé function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlevé equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function. We also notice that our work is a partial answer to one of the problems related to the β-ensembles formulated by Percy Deift during the June 2015 Montreal Conference on integrable systems. This paper is a contribution to the Special Issue on Asymptotics and Universality in Random Matrices,
 Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy.
 The full collection is available at http://www.emis.de/journals/SIGMA/Deift-Tracy.html. A. Its and T. Grava acknowledge the support of the Leverhulme Trust visiting Professorship
 grant VP2-2014-034. A. Its acknowledges the support by the NSF grant DMS-1361856 and by
 the SPbGU grant N 11.38.215.2014. A. Kapaev acknowledges the support by the SPbGU grant
 N 11.38.215.2014. F. Mezzadri was partially supported by the EPSRC grant no. EP/L010305/1.
 T. Grava acknowledges the support by the Leverhulme Trust Research Fellowship RF-2015-442
 from UK and PRIN Grant “Geometric and analytic theory of Hamiltonian systems in finite and
 infinite dimensions” of Italian Ministry of Universities and Researches. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Tracy-Widomβ Distribution for β=6 Article published earlier |
| spellingShingle | On the Tracy-Widomβ Distribution for β=6 Grava, T. Its, A. Kapaev, A. Mezzadri, F. |
| title | On the Tracy-Widomβ Distribution for β=6 |
| title_full | On the Tracy-Widomβ Distribution for β=6 |
| title_fullStr | On the Tracy-Widomβ Distribution for β=6 |
| title_full_unstemmed | On the Tracy-Widomβ Distribution for β=6 |
| title_short | On the Tracy-Widomβ Distribution for β=6 |
| title_sort | on the tracy-widomβ distribution for β=6 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148536 |
| work_keys_str_mv | AT gravat onthetracywidomβdistributionforβ6 AT itsa onthetracywidomβdistributionforβ6 AT kapaeva onthetracywidomβdistributionforβ6 AT mezzadrif onthetracywidomβdistributionforβ6 |