Vadim Gorin MIT (Cambridge) and IITP (Moscow) (based on joint … · 2017-06-06 · Central Limit...

Central Limit Theorem for discrete log�gases

Vadim GorinMIT (Cambridge) and IITP (Moscow)

(based on joint work with Alexei Borodin and Alice Guionnet)

June, 2015

Setup and overview

λ1 ≤ λ2 ≤ · · · ≤ λN , `i = λi + θi

Probability distributions on discrete N�tuples of the form.

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Discrete log�gas.We go beyond speci�c integrable weights.

• Appearance in probabilistic models of statistical mechanics.

• Law of Large Numbers and Central Limit Theorem for global�uctuations as N →∞ under mild assumptions on w(x ;N).

• Our main tool: discrete loop equations.

Setup and overview

λ1 ≤ λ2 ≤ · · · ≤ λN , `i = λi + θi

Probability distributions on discrete N�tuples of the form.

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Discrete log�gas.We go beyond speci�c integrable weights.

• Appearance in probabilistic models of statistical mechanics.

• Law of Large Numbers and Central Limit Theorem for global�uctuations as N →∞ under mild assumptions on w(x ;N).

• Our main tool: discrete loop equations.

Appearance of discrete log�gases

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

At θ = 1 becomes...

∏1≤i<j≤N

(`j − `i )2N∏i=1

w(`i ;N),

which frequently appears in natural stochastic systems.

Appearance of discrete log�gases

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

At θ = 1 becomes...

∏1≤i<j≤N

(`j − `i )2N∏i=1

w(`i ;N),

which frequently appears in natural stochastic systems.

Krawtchouk ensemble

• N independent simplerandom walks

• probability of jump p

• started at adjacent latticepoints

• conditioned never tocollide

Claim. (Konig�O'Connel�Roch) Distribution of N walkers at time t

∏1≤i<j≤N

(`j − `i )2N∏i=1

[p`i (1− p)M−`i

)], M = N + t − 1.

Claim. (Johansson) In random domino tilings of Aztec diamond.

Krawtchouk ensemble

• N independent simplerandom walks

• probability of jump p

• started at adjacent latticepoints

• conditioned never tocollide

∏1≤i<j≤N

(`j − `i )2N∏i=1

[p`i (1− p)M−`i

)], M = N + t − 1.

Krawtchouk ensemble

∏1≤i<j≤N

(`j − `i )2N∏i=1

[p`i (1− p)M−`i

)], M = N + t − 1.

Krawtchouk ensemble

∏1≤i<j≤N

(`j − `i )2N∏i=1

[p`i (1− p)M−`i

)], M = N + t − 1.

Hahn ensemble

• Regular A×B ×C hexagon

• 3 types of lozenges

• uniformly random tiling

• Distribution of N horizontallozenges on t�th vertical?

N = B + C − t

t > max(B,C )

(a)n = a(a+1) . . . (a+n−1)

Claim. (Cohn�Larsen�Propp)

∏i<j

(`i − `j)2N∏i=1

[(A + B + C + 1− t − `i )t−B (`i )t−C

Hahn ensemble

• 3 types of lozenges

N = B + C − t

t > max(B,C )

(a)n = a(a+1) . . . (a+n−1)

∏i<j

(`i − `j)2N∏i=1

[(A + B + C + 1− t − `i )t−B (`i )t−C

Hahn ensemble

N = B + C − t

t > max(B,C )

(a)n = a(a+1) . . . (a+n−1)

∏i<j

(`i − `j)2N∏i=1

[(A + B + C + 1− t − `i )t−B (`i )t−C

Hahn ensemble

N = B + C − t

t > max(B,C )

(a)n = a(a+1) . . . (a+n−1)

∏i<j

(`i − `j)2N∏i=1

[(A + B + C + 1− t − `i )t−B (`i )t−C

Two�interval support

• Rhombic hole of size D atvertical position H.

• Distribution of N horizontallozenges on the verticalgoing through the axis ofthe hole?

Claim. It is:(and similarly for k holes)

∏i<j

(`i−`j)2N∏i=1

[(A+B+C+1−t−ì )t−B (ì )t−C (H−ì )D (H−ì )D

∏i<j

(`i−`j)2N∏i=1

∏i<j

(`i−`j)2N∏i=1

∏i<j

(`i−`j)2N∏i=1

General θ case

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

• `i = L · xi , L→∞, β = 2θ.

∏1≤i<j≤N

(xj − xi )β

N∏i=1

w(`i ;N).

Eigenvalue ensembles of random matrix theory.β = 1, 2, 4 corresponds to real/complex/quaternion matrices.

• Another appearance � asymptotic representation theory

(Olshanski: (z,w)-measures).

FactorΓ(`j−ì+1)Γ(`j−ì+θ)Γ(`j−ì )Γ(`j−ì+1−θ) links to evaluation formulas for

Jack symmetric polynomials.

General θ case

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

• `i = L · xi , L→∞, β = 2θ.

∏1≤i<j≤N

(xj − xi )β

N∏i=1

w(`i ;N).

Eigenvalue ensembles of random matrix theory.β = 1, 2, 4 corresponds to real/complex/quaternion matrices.

• Another appearance � asymptotic representation theory

(Olshanski: (z,w)-measures).

FactorΓ(`j−ì+1)Γ(`j−ì+θ)Γ(`j−ì )Γ(`j−ì+1−θ) links to evaluation formulas for

Jack symmetric polynomials.

Large N setup

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

k regions with prescribed �lling fractions

n1 particles nk particles. . .

a1 b1 ak bk

1. w(·;N) vanishes at the boundaries of the regions.2. All data regularly depends on N →∞

ai = αiN + . . . , bi = βiN + . . . , ni = n̂iN + . . .

w(x ;N) = exp(NVN

)), NVN(z) = NV (z) + . . .

Potential V (z) should have bounded derivative(except at end�points, where we allow V (z) ≈ c · z ln(z)).

Large N setup

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

a1 b1 ak bk

1. w(·;N) vanishes at the boundaries of the regions.2. All data regularly depends on N →∞

ai = αiN + . . . , bi = βiN + . . . , ni = n̂iN + . . .

w(x ;N) = exp(NVN

)), NVN(z) = NV (z) + . . .

Potential V (z) should have bounded derivative(except at end�points, where we allow V (z) ≈ c · z ln(z)).

Law of Large Numbers

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Theorem. Suppose that all data regularly depends on N →∞,then the LLN holds: There exists µ(x)dx with 0 ≤ µ(x) ≤ θ−1,such that for any Lipshitz f and any ε > 0

limN→∞

N1/2−ε

∣∣∣∣∣ 1

N∑i=1

)−∫

f (x)µ(x)dx

∣∣∣∣∣ = 0

In fact the di�erence is O(1/N).

µ(x)dx is the unique maximizer of the functional IV

IV [ρ] = θ

∫∫x 6=y

ln |x − y |ρ(dx)ρ(dy)−∫ ∞−∞

V (x)ρ(dx).

This is a very general statement. Lots of analogues.

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

limN→∞

N1/2−ε

∣∣∣∣∣ 1

N∑i=1

)−∫

f (x)µ(x)dx

∣∣∣∣∣ = 0

IV [ρ] = θ

∫∫x 6=y

ln |x − y |ρ(dx)ρ(dy)−∫ ∞−∞

V (x)ρ(dx).

in appropriate class of measures taking into account �lling fractions

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

limN→∞

N1/2−ε

∣∣∣∣∣ 1

N∑i=1

)−∫

f (x)µ(x)dx

∣∣∣∣∣ = 0

IV [ρ] = θ

∫∫x 6=y

ln |x − y |ρ(dx)ρ(dy)−∫ ∞−∞

V (x)ρ(dx).

Law of Large Numbers: example

(Pictures by L. Petrov)

Graph of λi = `i − i (green lozenges) along the middle vertical

(Pictures by L. Petrov)

Graph of λi = `i − i (green lozenges) along the vertical axis of hole

The �lling fractions above and below the hole are �xed.

Averaged λi = `i − i (green lozenges) along the vertical axis of hole

• Frozen region: void. No particles, µ(x) = 0.• Frozen region: saturation. Dense packing, µ(x) = θ−1.• Band.

Central Limit Theorem?

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Is there a next order, as in CLT?

limN→∞

N∑i=1

)− Ef

• In continuous setting of RMT theory � yes, CLT.• Discreteness of the model might show up somewhere. E.g.local limits must be di�erent. Also there is rounding in 1/Nexpansion of E

∑f (`i/N). Can CLT feel being discrete?

• (Kenyon�2006), (Petrov�2012) CLT (GFF) for tilings of somesimply�connected domains. What if there are holes?

• Several other discrete CLT's exploit speci�c integrability.Methods not suitable for generic models. Approach ofJohansson seems to miss a critical ingredient in discrete world.

∏1≤i<j≤N

(xj − xi )β

N∏i=1

exp(−NV (xi ))

limN→∞

N∑i=1

)− Ef

• In continuous setting of RMT theory � yes, CLT.(Johansson�1998) one cut/one band, quite general V (x).. . .(Borot�Guionnet�2013) generic analytic V (x), �xed �llingfractions in each band. If not �xed ⇒ discrete component.

• Discreteness of the model might show up somewhere. E.g.local limits must be di�erent. Also there is rounding in 1/Nexpansion of E

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

limN→∞

N∑i=1

)− Ef

limN→∞

N∑i=1

)− Ef

Several other discrete CLT's exploit speci�c integrability.Methods not suitable for generic models. Approach of

Johansson seems to miss a critical ingredient in discrete world.

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

limN→∞

N∑i=1

)− Ef

• In continuous setting of RMT theory � yes, CLT.

• Discreteness of the model might show up somewhere. E.g.local limits must be di�erent. Also there is rounding in 1/Nexpansion of E

Central Limit Theorem

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

a1 b1 ak bk

Theorem. Assume that w(·;N) and V (·) are analytic (x ln(x)behavior of V at end�points is ok), all data depends on N regularly,and µ(x)dx is such that there is one band in each region. Thenunder technical assumptions, for analytic f1(x), . . . , fm(x)

limN→∞

N∑i=1

)− Efj

)], j = 1, . . . ,m.

are jointly Gaussian with explicit covariance.

Central Limit TheoremTheorem. Assume that all data depends on N regularly, V (x) isanalytic (expect for possible x ln(x) behavior at end�points), andµ(x)dx is such that there is one band in each region. Then undertechnical assumptions, for analytic f1(x), . . . , fm(x)

limN→∞

N∑i=1

)− Efj

)], j = 1, . . . ,m.

• In all the examples shown so far the technical assumption iseasy to check. Always holds for convex V (x) with one band.

• Conjecture (work in progress). Technical assumption holdsin generic case (e.g. a.s. in θ).

• The covariance depends only on end�points of the bands. Alog�correlated (generalized) Gaussian �eld. Section of 2d GFF.

• The result coincides with universal behavior in randommatrices / continuous β log�gases. (Johansson),(Bonnet�David�Eynard; Scherbina; Borot�Guionnet).

limN→∞

N∑i=1

)− Efj

)], j = 1, . . . ,m.

limN→∞

N∑i=1

)− Efj

)], j = 1, . . . ,m.

Central Limit Theorem

Theorem. Assume that all data depends on N regularly, V (x) isanalytic (expect for possible x ln(x) behavior at end�points), andµ(x)dx is such that there is one band in each region. Then undertechnical assumptions, for analytic f1(x), . . . , fm(x)

limN→∞

N∑i=1

)− Efj

)], j = 1, . . . ,m.

• For a number of particular models the result was establishedbefore.

• However this is the �rst generic result.(For θ = 1 case cf. the talk of Maurice Duits tomorrow)

Central Limit Theorem: example

Graph of `i − E`i (green lozenges) along the vertical axis of hole

The rough �uctuations aresmoothed in CLT

• The �lling fractions above and below the hole are �xed.• Comparison with RMT predicts that if we do not �x them,then a discrete component would appear.

Why?• Jump of one particle through the hole leads to a macroscopic�uctuation of

∑Ni=1 [f (`i/N)− Ef (`i/N)]

Graph of `i − E`i (green lozenges) along the vertical axis of hole

• Comparison with RMT predicts that if we do not �x them,then a discrete component would appear. Why?

• Jump of one particle through the hole leads to a macroscopic�uctuation of

∑Ni=1 [f (`i/N)− Ef (`i/N)]

• Comparison with RMT predicts that if we do not �x them,then a discrete component would appear. Why?

• Jump of one particle through the hole leads to a macroscopic�uctuation of

∑Ni=1 [f (`i/N)− Ef (`i/N)]

Form of measure

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

What's so special about this measure? Why not∏

i<j(`j − `i )β?

Recall: Johansson's CLT in RMT is based on loop equation

∏1≤i<j≤N

|xj − xi |βN∏i=1

exp(−NV (xi )).

GN(z) =1

N∑i=1

z − xi.

[EGN(z)

βV ′(z)

[EGN(z)

]+ (analytic) =

N(. . . )

Obtained by clever integration by parts.

Form of measure

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

i<j(`j − `i )β?

∏1≤i<j≤N

exp(−NV (xi )).

GN(z) =1

N∑i=1

z − xi.

[EGN(z)

βV ′(z)

[EGN(z)

]+ (analytic) =

N(. . . )

Obtained by clever integration by parts.

Form of measure

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

i<j(`j − `i )β?

GN(z)2 +2

βV ′(z)GN(z) + (analytic) =

N(. . . )

It also has applications far beyond. E.g. recently in edge universalityin RMT (Bourgade�Erdos�Yau), (Bekerman�Figalli�Guionnet)

Discrete CLT was long blocked by absence of a discrete analogue.

Form of discrete measure, for which an analogue could exist?

Can be hinted by discrete Selberg integrals.

`i = λi + (i − 1)θ, 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λN � integers∑∏i<j

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

Γ(`i + 1).

is explicit for all θ > 0 via Jack polynomials (+2 �binomial� w(x)).

Form of measure

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

i<j(`j − `i )β?

GN(z)2 +2

N(. . . )

∏1≤i<j≤N

w(x), w(x) =

xa(1− x)b 10<x<1,

xae−x 1x>0,

e−x2.

Known explicit formula manifests integrability of β log�gases.

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

Γ(`i + 1).

Form of measure

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

i<j(`j − `i )β?

GN(z)2 +2

N(. . . )

∏1≤i<j≤N

|xj−xi |βN∏i=1

w(x), w(x) =

px(1− p)M−x

)10≤x≤M ,

(x)M qx 1x≥0,

cx/x! 1x≥0.

Is known only at β = 2, but...

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

Γ(`i + 1).

Form of measure

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

i<j(`j − `i )β?

GN(z)2 +2

N(. . . )

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

Γ(`i + 1).

Main tool: Nekrasov equation

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Theorem. Assume

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(x), for analytic φ±N .

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

is analytic in the D ⊂ C, where φ±N are.

• This is a modi�cation of (Nekrasov�Pestun), (Nekrasov�Shatashvili), (Nekrasov)

• Knowing the statement, the proof is elementary.

• Discrete analogue of loop / Schwinger�Dyson equations.

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Theorem. Assume

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Theorem. Assume

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Theorem. Assume

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

Theorem. Assume

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

is analytic in D ⊂ C, where φ±N are.

How to use this theorem for asymptotic study?

• φ± � small degree polynomials (linear?), then the result isalso a polynomial. Find it to get equations.

• As degree grows, not very helpful. Need another approach.

Functions Rµ and Qµ

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

Regularity of data as N →∞ includes and implies

φ±N(Nz) = φ±(z) + . . . ,φ+(z)

φ−(z)= exp

(− ∂

∂zV (z)

Then ξ = Nz , N →∞ leads to analyticity of

Rµ(z) = φ−(z) exp(−θGµ(z)

)+ φ+(z) exp

(θGµ(z)

)Gµ is the Stieltjes transform of limiting density.

Gµ(z) =

z − xµ(x)dx .

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

Regularity of data as N →∞ includes and implies

φ±N(Nz) = φ±(z) + . . . ,φ+(z)

φ−(z)= exp

(− ∂

∂zV (z)

)Then ξ = Nz , N →∞ leads to analyticity of

)+ φ+(z) exp

(θGµ(z)

Gµ(z) =

z − xµ(x)dx .

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

Then ξ = Nz , N →∞ leads to analyticity of

)+ φ+(z) exp

(θGµ(z)

)We also need

Qµ(z) = φ−(z) exp(−θGµ(z)

)− φ+(z) exp

(θGµ(z)

Gµ(z) =

z − xµ(x)dx .

Gµ(z) =

z − xµ(x)dx .

)+ φ+(z) exp

(θGµ(z)

)Qµ(z) = φ−(z) exp

(−θGµ(z)

)− φ+(z) exp

(θGµ(z)

)n1 particles nk particles. . .

a1 b1 ak bk

Key technical assumption: for analytic H(z)

Qµ(z) = H(z)k∏

√(z − ui )(z − vi ), H(z) 6= 0.

• Quadratic singularities: Qµ(z) =√

Rµ(z)2 − 4φ+(z)φ−(z).

• ui and vi must be end�points of bands.

Gµ(z) =

z − xµ(x)dx .

)+ φ+(z) exp

(θGµ(z)

(−θGµ(z)

)− φ+(z) exp

(θGµ(z)

)n1 particles nk particles. . .

a1 b1 ak bk

Key technical assumption: for analytic H(z)

Qµ(z) = H(z)k∏

√(z − ui )(z − vi ), H(z) 6= 0.

• Quadratic singularities: Qµ(z) =√

Rµ(z)2 − 4φ+(z)φ−(z).

• ui and vi must be end�points of bands.

Second order expansion

φ−N(ξ) ·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ) ·E

[N∏i=1

ξ − `i − 1

)+ φ+(z) exp

(θGµ(z)

(−θGµ(z)

)− φ+(z) exp

(θGµ(z)

)Second order expansion as N →∞ gives

Qµ(z) · NE(GN(z)− Gµ(z)) = (explicit) + (analytic) + (small).

Here Gµ(z) =

z − xµ(x)dx , GN(z) =

N∑i=1

z − `i/N.

(small) requires non-trivial technical work

Gµ(z) =

N∑i=1

z − `i/N.

Second order expansion as N →∞ gives

H(z)k∏

√(z − ui )(z − vi ) · NE(GN(z)− Gµ(z))

= (explicit) + (analytic) + (small).

z − y

k∏i=1

√(z − ui )(z − vi ) · NE(GN(z)− Gµ(z))

Integrate aroundk⋃

i=1[ui , vi ] to get lim

N→∞NE(GN(y)− Gµ(y)).

• We use one band per interval, as otherwise we can notintegrate due to singularities of GN .

• We use �xed �lling fractions, to resolve the contribution ofthe residue at ∞.

• We use H(z) 6= 0, as otherwise the unknown (analytic) wouldcontribute.

Gµ(z) =

N∑i=1

z − `i/N.

Second order expansion as N →∞ gives

H(z)k∏

√(z − ui )(z − vi ) · NE(GN(z)− Gµ(z))

z − y

k∏i=1

√(z − ui )(z − vi ) · NE(GN(z)− Gµ(z))

z − y

k∏i=1

√(z − ui )(z − vi ) · NE(GN(z)− Gµ(z))

Proof of Central Limit Theorem

Gµ(z) =

N∑i=1

z − `i/N.

We explicitly found limN→∞

NE(GN(y)− Gµ(y)).

Proposition. Deform the weight by m factors

w(x ;N)→ w(x ;N)m∏

taya − x/N

Then limN→∞ of the mixed ta derivative at 0 ofNE(GN(y)− Gµ(y)) gives joint cumulants of

NE(GN(y)− Gµ(y)), NE(GN(ya)− Gµ(ya)), a = 1, . . .m.

The deformed measure is in the same class. If we justify

interchange of derivation and N →∞ limit, then the cumulantsyield asymptotic Gaussianity and the expression for covariance.

Gµ(z) =

N∑i=1

z − `i/N.

w(x ;N)→ w(x ;N)m∏

taya − x/N

Gµ(z) =

N∑i=1

z − `i/N.

w(x ;N)→ w(x ;N)m∏

taya − x/N

Gµ(z) =

N∑i=1

z − `i/N.

w(x ;N)→ w(x ;N)m∏

taya − x/N

Then limN→∞ of mixed ta derivative at 0 of NE(GN(y)− Gµ(y))gives joint cumulants of NE(GN(ya)− Gµ(ya))

Result: limNE(GN(y)− EGN(y)) � Gaussian. One band [u, v ] :

limN→∞

N2E[GN(y)GN(z)− EGN(y)EGN(z)

]= − 1

2(y − z)2

yz − 12 (u + v)(y + z) + u + v√

(y − u)(y − v)√

(z − u)(z − v)

An explicit integral expression for k bands.

Summary

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

1. Central limit theorem with universal covariance under• One band per interval of support.• Technical assumption, which holds in many cases, e.g.

(z ,w)�measures of asymptotic representation theoryw(x ;N) = exp(NV (x/N)) with convex VConjecture (work in progress). In generic situation.

Summary

∏1≤i<j≤N

Γ(`j − `i + 1)Γ(`j − `i + θ)

Γ(`j − `i )Γ(`j − `i + 1− θ)

N∏i=1

w(`i ;N),

1. Central limit theorem with universal covariance under• One band per interval of support.• Technical assumption, which holds in many cases.

Conjecture (work in progress). In generic situation.

2. An important ingredient of the proof is Nekrasov equation( discrete loop / Schwinger�Dyson equation )

w(x ;N)

w(x − 1;N)=φ+N(x)

φ−N(ξ)·E

[N∏i=1

(1− θ

ξ − `i

)]+φ+

N(ξ)·E

[N∏i=1

ξ − `i − 1

)]is analytic in D ⊂ C, where φ±N are.

Vadim Gorin MIT (Cambridge) and IITP (Moscow) (based on joint … · 2017-06-06 · Central Limit...

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