Painlevé Equations: Analysis and Applications

Painlevé Equations: Analysis andApplications

Lun ZhangSchool of Mathematical Sciences, Fudan University

2 | 36

Outline of the talk

I Part I – Location of poles for the Hastings-McLeod solution tothe Painlevé II equation

Joint work with Min Huang and Shuai-Xia Xu

I Part II – Gap probability at the hard edge for random matrixensembles with pole singularities in the potential

Joint work with Dan Dai and Shuai-Xia Xu

Painlevé Equations: Analysis and Applications

Part I – Introduction 3 | 36

Full list of Painlevé equationsd2wdz2 = 6w2 + z

d2wdz2 = 2w3 + zw + α

d2wdz2 =

1w

(dwdz

)2

− 1z

dwdz

+αw2 + β

z+ γw3 +

δ

w

d2wdz2 =

12w

(dwdz

)2

+32w3 + 4zw2 + 2(z2 − α)w +

β

w

d2wdz2 =

(1

2w+

1w − 1

)(dwdz

)2

−1z

dwdz

+(w − 1)2

z2

(αw +

β

w

)+γwz

+δw(w + 1)

w − 1Painlevé VI . . .



A short history of Painlevé equations

I The Painlevé equations possess the so-called Painlevé property:all the solutions are free from movable branch points

I Discovered by Painlevé and his colleagues at the beginning of20th century while classifying all second-order ordinarydifferential equations

d2wdz2 = R(z ,w ,

dwdz

),

which possess the Painlevé property

I The solutions of Painlevé equations are called the Painlevétranscendents






d2wdz2 = R(z ,w ,

dwdz

),








d2wdz2 = R(z ,w ,

dwdz

),





n-truncated solutions

I The first two Painlevé equations

y ′′ = 6y2 + x , y ′′ = 2y3 + xy + α

All solutions are meromorphic x = ∞ is the only essential singularity

I Existence of solutions which have no lines of poles near infinitynear n (n = 1, 2, 3) of the critical rays – n-truncated solutions

[Boutroux, 1913&1914]




I The first two Painlevé equations

y ′′ = 6y2 + x , y ′′ = 2y3 + xy + α

All solutions are meromorphic x = ∞ is the only essential singularity

I Existence of solutions which have no lines of poles near infinitynear n (n = 1, 2, 3) of the critical rays – n-truncated solutions

[Boutroux, 1913&1914]




I Critical rays:

Γk :=

x∣∣∣ arg x =

2kπN

, k = 0, 1, . . . ,N − 1,

where

N =

5, for PI6, for PII

I Examples:




I Critical rays:

Γk :=

x∣∣∣ arg x =

2kπN

, k = 0, 1, . . . ,N − 1,

where

N =

5, for PI6, for PII

I Examples:



A conjecture of Novokshenov

Conjecture (Novokshenov, ’14)If the 2- or 3-truncated solution of Painlevé equation has no pole atinfinity in a sector Ξk , then it has no poles in the whole sector Ξk ,where

Ξk :=

x∣∣∣ 2kπ

N< arg x <

2(k + 1)πN

, k = 0, 1, . . . ,N − 1.

I Numerical confirmations:[Fornberg-Weideman, ’11&’14; Novokshenov ’09]




I For the 3-truncated solutions (tritronquées) of PI: Dubrovin’sconjecture

[Dubrovin-Grava-Klein, ’09]

I The tritronquée solution describes the asymptotic behavior ofsolutions to the focusing NLS equation near the critical point.Poles of the tritronquées are related to the spikes of NLSsolutions

[Bertola-Tovbis, ’13]

I Dubrovin’s conjecture has been completely proved recently[Costin-Huang-Tanveer, ’14]



















Hastings-McLeod solution of PII

I The Hastings-McLeod solution yHM is a special solution of

y ′′ = 2y3 + xy

with the asymptotics

yHM(x) ∼

Ai(x), as x → +∞√−x/2, as x → −∞

I The solution yHM is known to be pole-free on the real axis[Hastings-McLeod, ’80]



Hastings-McLeod solution of PII

I The Hastings-McLeod solution yHM is a special solution of

y ′′ = 2y3 + xy

with the asymptotics

yHM(x) ∼

Ai(x), as x → +∞√−x/2, as x → −∞

I The solution yHM is known to be pole-free on the real axis[Hastings-McLeod, ’80]



Tracy-Widom distribution in RMT

I For the largest eigenvalue λmax of an n × n GUE matrix, therandom variable

n1/6(λmax − 2√

n)

converges in distribution to the well-known Tracy-Widomdistribution F2(s) as n → ∞

[Tracy-Widom, ’94]

I Tracy-Widom distribution is universal random permutation

[Baik-Deift-Johansson, ’99] Asymmetric Simple Exclusion Process (ASEP) with step initial

condition[Johansson, ’00; Tracy-Widom, ’09]

· · · · · ·



Tracy-Widom distribution in RMTI For the largest eigenvalue λmax of an n × n GUE matrix, the

random variablen1/6(λmax − 2

√n)

converges in distribution to the well-known Tracy-Widomdistribution F2(s) as n → ∞

[Tracy-Widom, ’94]

I Tracy-Widom distribution is universal random permutation

[Baik-Deift-Johansson, ’99] Asymmetric Simple Exclusion Process (ASEP) with step initial

condition[Johansson, ’00; Tracy-Widom, ’09]

· · · · · ·



Tracy-Widom distribution and yHM

I There are two formulas for the Tracy-Widom distribution

Fredholm determinant representation:

F2(s) = det(I − As)

where As is the integral operator acting on L2(s,∞) withkernel given in terms of Airy functions Ai by

Ai(x)Ai′(y)− Ai′(x)Ai(y)x − y

Airy kernel

Integral representation:

F2(s) = exp(−∫ ∞

s(x − s)y2

HM(x) dx)



Tracy-Widom distribution and yHM

I There are two formulas for the Tracy-Widom distribution Fredholm determinant representation:

F2(s) = det(I − As)

where As is the integral operator acting on L2(s,∞) withkernel given in terms of Airy functions Ai by

Ai(x)Ai′(y)− Ai′(x)Ai(y)x − y

Airy kernel

Integral representation:

F2(s) = exp(−∫ ∞

s(x − s)y2

HM(x) dx)


Part I – Statement of results 12 | 36

The Hastings-MeLeod solution and its poles



The Hastings-MeLeod solution and its poles



Main result

Theorem (Huang-Xu-LZ, Constr. Approx., ’16)The Hastings-McLeod solution yHM of the second homogeneousPainlevé equation

y ′′ = 2y3 + xy

is pole-free in the region arg x ∈ [−π3 ,

π3 ] ∪ [2π3 ,

4π3 ].



Known results

I For |x | large enough – Riemann-Hilbert approach[Its-Kapaev, ’03]

I For arg x ∈ [−π3 ,

π3 ] – an operator-norm estimate

[Bertola, ’12]


Part I – About the proof 15 | 36

Strategy of proof

I A direct analysis based on the idea in the proof of Dubrovin’sconjecture

I Construct an explicit quasi-solution and show the differencebetween real solution and the quasi-solution is small in asuitable norm Difficulty: an effective quasi-solution approximation with

sufficient accuracy for both small and large |x |

I Can be applied to other equations including the general PIIequation with α = 0



Strategy of proof







Strategy of proof







Strategy of proof

I Focus on the sector

Ω :=

x ∈ C∣∣∣ 2π/3 ≤ arg x ≤ π

I Analyze yHM in two regions

Ω0 :=

x ∈ C

∣∣∣ |x | > 34/3

2, 2π/3 6 arg x 6 π

and

Ω2 :=

x ∈ C∣∣∣ |x | 6 9/4, 2π/3 6 arg x 6 π

.

I Note that Ω ⊆ Ω0 ∪ Ω2



Properties of yHM

Proposition (Its-Kapaev, ’03)Let yHM be the Hastings-McLeod solution of the second Painlevéequation, then

yHM(x) =1

2√π

x−1/4e−23 x3/2

(1 +O(x−3/4)

)as x → +∞ and arg x = 0;

yHM(x) =√

−x/2(1 +O((−x)−3/2)

)+ c−(−x)−1/4e−

2√

23 (−x)3/2

(1 +O(x−1/4)

)as x → ∞ and arg x ∈ [2π3 ,

4π3 ), where c− = i2−7/4

√π

.Painlevé Equations: Analysis and Applications


Analysis of yHM in the region Ω0

I Change of variables:

t =23

√2(−x)3/2, y(x) =

3√

3t2

h(t),

hence,

yHM 7→ hHM , Ω0 7→ Ω1 :=

t ∈ C∣∣∣ |t| > 3,−π/2 6 arg t 6 0

I We have hHM(t) = hp(t) + he(t), where

hp(t) = 1 − 19t2 +

h2(t)t4 ∼ 1 − 1

9t2 , |h2(t)| 665,

he(t) =ce−t√

t(ha(t) + δ1(t)) ∼

√2ce−t√

t,

with ha being a quasi-solution and |δ1(t)| 6 52|t|2



Analysis of yHM in the region Ω0

I Change of variables:

t =23

√2(−x)3/2, y(x) =

3√

3t2

h(t),

hence,

yHM 7→ hHM , Ω0 7→ Ω1 :=

t ∈ C∣∣∣ |t| > 3,−π/2 6 arg t 6 0

I We have hHM(t) = hp(t) + he(t), where

hp(t) = 1 − 19t2 +

h2(t)t4 ∼ 1 − 1

9t2 , |h2(t)| 665,

he(t) =ce−t√

t(ha(t) + δ1(t)) ∼

√2ce−t√

t,

with ha being a quasi-solution and |δ1(t)| 6 52|t|2



Analysis of yHM in the finite region Ω2

I When |x | becomes small, no asymptotic expansion can providesufficient information about yHM and the initial values at afinite point are needed

I To get approximations of initial values at the origin withcontrolled error bound Analysis of yHM for x > 3:

quasi-solution → aproximations of yHM(3) and y ′HM(3)

Analysis of yHM for 0 ≤ x ≤ 3:





I When |x | becomes small, no asymptotic expansion can providesufficient information about yHM and the initial values at afinite point are needed

I To get approximations of initial values at the origin withcontrolled error bound Analysis of yHM for x > 3:


Analysis of yHM for 0 ≤ x ≤ 3:





I From approximations of initial values at the origin, we obtain

|yHM(x)− yb(x)| < 6/5, x ∈ Ω2,

with yb being the quasi-solution (a polynomial of degree 15)

I Technical parts: Contractive map in a suitable Banach space Taylor series / fitting numerical data Estimating a real/complex polynomial over an interval/ a

domain




I From approximations of initial values at the origin, we obtain

|yHM(x)− yb(x)| < 6/5, x ∈ Ω2,

with yb being the quasi-solution (a polynomial of degree 15)

I Technical parts: Contractive map in a suitable Banach space Taylor series / fitting numerical data Estimating a real/complex polynomial over an interval/ a

domain


21 | 36

Part II – Gap probability at the hard edge for randommatrix ensembles with pole singularities in the

potential


Part II – Introduction 22 | 36

The model

I A probability measure

1Zn

(det M)α exp[−ntr Vk(M)] dM, α > −1,

defined on the space of n × n positive definite Hermitianmatrices where Zn: a normalization constant dM: flat complex Lebesgue measures on the entries the potential

Vk(x) := V (x) +( t

x

)k, x ∈ (0,∞), t > 0



Eigenvalue distribution

I Joint probability density function of the eigenvalue distribution:

1Cn

∏1≤i<j≤n

(xj − xi )2

n∏j=1

w(xj),

wherew(x) = xαe−nVk(x)

I Correlation kernel

Kn(x , y ; t) = h−1n−1

√w(x)w(y)

πn(x)πn−1(y)− πn−1(x)πn(y)x − y

,

where ∫ ∞

0πj(x)πm(x)w(x) dx = hjδj ,m



Eigenvalue distribution

I Joint probability density function of the eigenvalue distribution:

1Cn

∏1≤i<j≤n

(xj − xi )2

n∏j=1

w(xj),

wherew(x) = xαe−nVk(x)

I Correlation kernel

Kn(x , y ; t) = h−1n−1

√w(x)w(y)

πn(x)πn−1(y)− πn−1(x)πn(y)x − y

,

where ∫ ∞

0πj(x)πm(x)w(x) dx = hjδj ,m



Motivations

I Statistics for zeta zeros and eigenvalues – probabilitydistribution of Tuck’s function

[Berry-Shukla, ’08]

I Quantum transport and electrical characteristics of chaoticcavities – eigenvalues of Wigner-Smith time-delay matrix

[Brouwer-Frahm-Beenakker, ’97&’99; Grabsch–Texier, ’14]

I The field of spin-glasses – random matrix model arising inmean-field glassy systems

[Akemann-Villamaina-Vivo,’14]



Recent progresses

I Singularly perturbed GUE: Vk(x) = 12x2 + t

2x2 , α = 0 Double scaling limit of the partition function – related to the

Painlevé III equation[Mezzadri-Mo, ’09; Brightmore-Mezzadri-Mo,’15]

I Singularly perturbed LUE: Vk(x) = x + tx

Connection with the Painlevé III equation for finite n[Chen-Its,’10]

Double scaling limit for the correlation kernel – model RHproblem associated to the Painlevé III equation

[Xu-Dai-Zhao,’14]

I General potential: Vk(x) = V (x) +( t

x

)k

Connection with a Painlevé III hierarchy[Atkin-Claeys-Mezzadri, ’16]



Recent progresses







[Xu-Dai-Zhao,’14]


x

)k




Recent progresses







[Xu-Dai-Zhao,’14]


x

)k




A Riemann-Hilbert (RH) problem

(1) Ψ(z ;λ) is analytic in C \ ∪3j=1Σj .

(2) Jump conditions:

JJ

JJ

JJ

J

^

-

0

Σ3

Σ1

Σ2

(0 1−1 0

)(

1 0eαπi 1

)

(1 0

e−απi 1

)

r



A Riemann-Hilbert (RH) problem

(1) Ψ(z ;λ) is analytic in C \ ∪3j=1Σj .

(2) Jump conditions:

JJ

JJ

JJ

J

^

-

0

Σ3

Σ1

Σ2

(0 1−1 0

)(

1 0eαπi 1

)

(1 0

e−απi 1

)r



RH problem for Ψ

(3) As z → ∞,

Ψ(z ;λ) =

I +

(q(λ) −ir(λ)ip(λ) −q(λ)

)z

+O(z−2)

× z−

14σ3

I + iσ1√2

e√

zσ3

(4) As z → 0,

Ψ(z ;λ) = Ψ0(λ)(I +O(z))e−(−λz )

kσ3z

α2 σ3Hj



Remarks about the RH problem

I This RH problem is uniquely solvable for k ∈ N, α > −1 andλ > 0

[Xu-Dai-Zhao,’14; Atkin-Claeys-Mezzadri,’16]

I λ = 0: explicitly solvable in terms of modified Bessel functions[Kuijlaars-McLaughlin-Van Assche-Vanlessen,’04]

I Connection with a Painlevé III hierarchy: k + 1 ODEs for k + 1unknown functions (ρ(λ), ℓ1(λ), . . . , ℓk(λ))

ρ = − 14ℓ2k

((ℓ2k)′′ − 3(ℓ′k)

2 + τ0), p = 0,p∑

q=0(ℓk−p+q+1ℓk−q − (ℓk−p+qℓk−q)

′′+

3ℓ′k−p+qℓ′k−q − 4ρℓk−p+qℓk−q) = τp, 1 ≤ p ≤ k





[Xu-Dai-Zhao,’14; Atkin-Claeys-Mezzadri,’16]I λ = 0: explicitly solvable in terms of modified Bessel functions

[Kuijlaars-McLaughlin-Van Assche-Vanlessen,’04]

I Connection with a Painlevé III hierarchy: k + 1 ODEs for k + 1unknown functions (ρ(λ), ℓ1(λ), . . . , ℓk(λ))

ρ = − 14ℓ2k

((ℓ2k)′′ − 3(ℓ′k)

2 + τ0), p = 0,p∑

q=0(ℓk−p+q+1ℓk−q − (ℓk−p+qℓk−q)

′′+






[Xu-Dai-Zhao,’14; Atkin-Claeys-Mezzadri,’16]I λ = 0: explicitly solvable in terms of modified Bessel functions

[Kuijlaars-McLaughlin-Van Assche-Vanlessen,’04]I Connection with a Painlevé III hierarchy: k + 1 ODEs for k + 1

unknown functions (ρ(λ), ℓ1(λ), . . . , ℓk(λ))ρ = − 1

4ℓ2k((ℓ2k)

′′ − 3(ℓ′k)2 + τ0), p = 0,

p∑q=0

(ℓk−p+q+1ℓk−q − (ℓk−p+qℓk−q)′′+





Proposition (Atkin-Claeys-Mezzadri,’16)Let

yα(λ) := −2iddλ

(r(λ2)

).

Then, yα(λ) is a solution of the equation for ℓ1 of the k-th memberof the aforementioned Painlevé III hierarchy with

τp =

42k+1k2, p = 0,−(−4)k+1αk, p = k,0, 0 < p < k

In addition, the asymptotics of r , hence yα is known.



Double scaling limit of Kn at the hard edge

I(ψ1(z ;λ), ψ2(z ;λ)

)t : analytic extension of first column ofΨ(z ;λ) in the region bounded by Γ1 and Γ3

I If t → 0 and n → ∞ in such a way that2−

1k c1n

2k+1k t → λ > 0, then

limn→∞

1c1n2 Kn

(u

c1n2 ,v

c1n2 ; t)

= KPIII(u, v ;λ),

where

KPIII(u, v ;λ) := eαπi ψ1(−v ;λ)ψ2(−u;λ)− ψ1(−u;λ)ψ2(−v ;λ)2πi(u − v)




Double scaling limit of Kn at the hard edge

I(ψ1(z ;λ), ψ2(z ;λ)

)t : analytic extension of first column ofΨ(z ;λ) in the region bounded by Γ1 and Γ3

I If t → 0 and n → ∞ in such a way that2−

1k c1n

2k+1k t → λ > 0, then

limn→∞

1c1n2 Kn

(u

c1n2 ,v

c1n2 ; t)

= KPIII(u, v ;λ),

where

KPIII(u, v ;λ) := eαπi ψ1(−v ;λ)ψ2(−u;λ)− ψ1(−u;λ)ψ2(−v ;λ)2πi(u − v)



Part II – Main result 31 | 36

Main result

Theorem (Dai-Xu-LZ, ’17)Let KPIII be the integral operator with kernel KPIII(u, v)χ[0,s](v)acting on the function space L2((0,∞)). Then,

ln det(I −KPIII) = − s4+ αs1/2 − α2

4ln s

+

∫ λ

0

12t

(r(t) +

α2

2− 1

8

)dt + τα +O(s−1/2), s → +∞,

where r is related to a Painlevé III hierarchy, τα = ln(

G(1+α)

(2π)α/2

)with

G (z) being the Barnes G-function


Part II – About the proof and future work 32 | 36

About the proof

I Relies on the integrable (in the sense of IIKS) structure ofKPIII and a Deift/Zhou steepest analysis of the associated RHproblem

I Large s asymptotics of Fredholm determinant associated withother Painlevé kernels: Painlevé I hierarchy

[Claeys-Its-Krasovsky,’10] Painlevé II kernel (Hastings-McLeod solution)

[Bothner-Its,’14] Painlevé II kernel (Ablowitz-Segur solution)

[Bothner-Buckingham,’17] Painlevé XXXIV kernel

[Xu-Dai,’17]



About the proof

I Relies on the integrable (in the sense of IIKS) structure ofKPIII and a Deift/Zhou steepest analysis of the associated RHproblem

I Large s asymptotics of Fredholm determinant associated withother Painlevé kernels: Painlevé I hierarchy

[Claeys-Its-Krasovsky,’10] Painlevé II kernel (Hastings-McLeod solution)

[Bothner-Its,’14] Painlevé II kernel (Ablowitz-Segur solution)

[Bothner-Buckingham,’17] Painlevé XXXIV kernel

[Xu-Dai,’17]



About the proof

Step 1 Large s asymptotics of dds F (s;λ) with

F (s;λ) := ln det(I − KPIII)

Fact

dds

ln F (s;λ) = −eαπi

2πilim

z→−s

(X−1(z)X ′(z)

)21

Step 2 Large s asymptotics of ddλF (s;λ)

Factddλ

F (λ2s;λ2) = r(λ2)/λ− (X∞)12



About the proof

Step 1 Large s asymptotics of dds F (s;λ) with

F (s;λ) := ln det(I − KPIII)

Fact

dds

ln F (s;λ) = −eαπi

2πilim

z→−s

(X−1(z)X ′(z)

)21

Step 2 Large s asymptotics of ddλF (s;λ)

Factddλ

F (λ2s;λ2) = r(λ2)/λ− (X∞)12



About the proof

Step 3 The constant term Fact

KPIII(u, v ;λ) = KBes(u, v) +O(λ), λ→ 0,

where

KBes(x , y) =Jα(

√x)√

yJ ′α(√

y)−√

xJ ′α(√

x)Jα(√

y)2(x − y)

As s → +∞,

ln det(I−KBes) = −14s+αs1/2−α

2

4ln s+ln

(G (1 + α)

(2π)α/2

)+O(s−1/2)

[Deift-Krasovsky-Vasilevska,’11]



About the proof

Step 3 The constant term Fact

KPIII(u, v ;λ) = KBes(u, v) +O(λ), λ→ 0,

where

KBes(x , y) =Jα(

√x)√

yJ ′α(√

y)−√

xJ ′α(√

x)Jα(√

y)2(x − y)

As s → +∞,

ln det(I−KBes) = −14s+αs1/2−α

2

4ln s+ln

(G (1 + α)

(2π)α/2

)+O(s−1/2)

[Deift-Krasovsky-Vasilevska,’11]



Future work

I Tracy-Widom type formula for the gap probability?

Coupled Painlevé III system for k = 1



Thanks for your attention!


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Painlevé Equations: Analysis and Applications