Graphic Enumerations and Discrete Painleve
Equations via Random Matrix Models
Chuan-Tsung Chan
Department of Applied Physics, Tunghai University, Taiwan
based on the paper arXiv:1712.09231 with Dr. Hsiao-Fan Liu—————————————————————————————–• The free engergies (FE) of random matrix model (RMM) integralscorrespond to the generating functions for the enumeration of discrete random surfaces.
• We devise a new computer-programable scheme for an exact evaluation of the generating function based on a ”topological expansion” of
the recursive coefficients of the associated orthogonal polynomials.

Grand Scheme of this Work—————————————————————————————-

Graphic Enumerations

diagrammaticexpansion

topologicalexpansion

Random Matrix
ModelsDiscreteDiﬀerence Eqs.
orthogonal
polynomial

Tactics of Solutions—————————————————————————————-

diﬀerence equations(recurrence rel.)

generating
function

algebraic equations

Frobenius
methoddiﬀerential equations

Feynman Rules and Topological Expansion—————————————————————————————-

N dependence for a fat-graph⇒ topological expansion
weighting for a junction ⇒ N g,weighting for a propogator ⇒ 1N ,weighting for a loop ⇒ N ,dual transformation:n ⇒ F, P ⇒ E, L ⇒ V.
total weighting ⇒(N g)nN −P N L = g nN n−P +L= g nN F −E+V = g nN χ.

Orthogonal Polynomial Approach to RMM—————————————————————————————
R∞• Orthogonal polynomials : −∞ Pn(λ)Pm(λ) e−N V (λ) dλ = hn δmn.
• Partition functionof RMM in terms of normalization constants
ZZ (g, N )DM e−N trV (M )=HN=⇒|{z}

1N !
13!

N =3

⇒Z YNZdλk det Pj−1(λi)

k=1
3Ydλkk=1
NY−1

2

i,j−NePN
l=1

V (λl )

#2"Xσ(−1) P0(λσ(1))P1(λσ(2))P2(λσ(3))

e−3

P3
l=1

V (λl )σ∈π3hn(g, N ).

(1)

n=0

• Three-term recurssive relation :quartic model ⇒ λPn(λ) = Pn+1(λ) + rnPn−1(λ).
cubic model ⇒ λPn(λ) = Pn+1(λ) + βnPn(λ) + αnPn−1(λ).

(2)(3)

Discrete Painleve Equation from RMM————————————————————————————• hn =R∞−∞ Pn−1 (λ) [λPn (λ)]

e−N V (λ) dλ = rnhn−1 ⇒ rn = hnhn−1.

• Partition function for the RMM : Z (g, N ) = hN
0

QN −1
k=1

rkN −k .

• Recursive equation(s)among recurrence coefficientsZ∞

nhn−1 =−∞

Pn−1(λ) [ddλPn(λ)] e−N V (λ)dλ= Pn−1(λ)Pn(λ)e−N V (λ)∞−∞ −+ NZ ∞−∞Z ∞−∞(ddλPn−1) Pne−N V dλ0

V (λ)Pn−1(λ)Pn(λ)e−N V (λ)dλ(4)n= rn − grn (rn+1 + rn + rn−1) , d-P.I Eq. (5)Nn= αn[1 − g(βn + βn−1)],(6)
cubic model ⇒Nβn = g(αn+1 + βn2 + αn).
(7)quartic model ⇒1

Roadmap of the Computations (Part I)• Partition Function of the Random Matrix Models (RMM)
Z=N−1Yhl .

(8)

l=0

• Recursive coefficients of the associated monic orthogonal polynomial systems(for even potential V (−x) = V (x))
x · Pn (x) = Pn+1 (x) + rn Pn−1 (x),⇒ ln Z= N (ln h0 ) +N−1X

rk =(N − k) ln rk .

hk.
hk−1(9)(10)

k=1• Perturbative expansion of the recursive cofficientsfor the quartic model
∞
1 X
g
rn =
rnm m+1 ,  :=
g m=0
N

(11)

Roadmap of the Computations (Part II)• Polynomality of the Recursive Coefficientsrnm =m+1[X2 ](l)
nm+1−2l 3m−l Cm,

C (l) (x) :=∞X(l) mCmx .

(12)m=0l=0• Topological Expansion of the Recursive Coefficients∞rn (g, N ) ==n X(3n2 )−l C (l) (3n)N
13gl=0∞X(3n2 )−l R(l) (3n),(13)
(l)
lR(x)
:= xC(x)
.

(14)

l=0

• Solutions of the Topological Coefficients√∞X
1−
1 − 4x
(2n)!
(0)
C
=
=xn ,2x(n + 1)!n!
n=0√22xx(1−1 − 4x)(0)
C (1) =C(x)=,22
(1 − 4x)(1 − 4x)44−28x
98x
(0)
+C
(x)
C (2) =
49/2
(1 − 4x)(1 − 4)

(15)
(16)
(17)

Roadmap of the Computations (Part III)
• Free Energy of the Random Matrix ModelZ(g, N )F (g, N ) := lnZ(0, N ) NX
−1h0 (g, N )rk (g, N )= N ln+(N − k) lnh0 (0, N )rk (0, N )

(18)
(19)

k=1

=N 2 e0 (g) + N 0 e1 (g) + N −2 e2 (g) + O(N −4 )(20)

• Topological Expansion of the Finite N Matrix Modele0 (g) =∞X(2k − 1)!

(3g)k(k + 2)!k!
k=0∞ 1 X 4k(2k)!
e1 (g) =−(3g)k
24k(k + 1)k!

(21)

(22)

k=1∞

e2 (g) =X12k−3[195·2(k − 1)
25 · 33 · 5
k=3−(2k)!
(28k + 9)(h − 1)](3g)kk!

(23)

Summary & Outlook (Part I)———————————————————————————–• Take-home Message
Z (g, N ) :=Z

DM e−N trV (M )HN

F (g, N ) := ln [Z (g, N )Z (0, N )] = N2X

g n N −2h Cn,h.

n,hF (e−β , eγ ) =X

e−βn+γ(2−2h) Cn,hn,h• We have performed systematic computations for counting DRS :– perturbative expansion in g ⇒ F (g, N ) = N 2fn(N−2) :=PhN−2hPPngnnfn(N −2)

Cn,h finite (and even) polynomials in 1/N 2.

– topological expansion in N −2 ⇒ F (g, N ) = N 2eh(g) :=ng

Cn,h infinite series in g.

PhN−2heh(g)Summary & Outlook (Part II)—————————————————————————————• Our calculations are based on the solutions of discrete difference
equations as derived from the orthogonal polynomial approach to
the random matrix integrals.
• Throughout our computations, we did not take any continuous approximation of summation. Furthermore, the relevant datum are
obtained through the solutions of finite difference equations. Hence,
our computations provide independent check of previous results, andlay down the foundation for a rigorous study of continuous/nonperturbative limit of 2-d quantum gravity.
• Hopeful, our analysis of perturbative solutions to the non-linear difference equations may shed some light for further explorations (e.g.,asymptotic behaviors, exact solutions etc.)• Double-Scaling Limit (DSL):
generating function for DRS ⇒ partition function for minimal string
theory.