Table Of ContentRecursion relations for Unitary integrals,
Combinatorics and the Toeplitz Lattice
2
0
0
2
M. Adler P. van Moerbeke
∗ †
n
a
J January 25, 2002
0
3
1
v
3
Contents
6
0
1
0 Introduction and main results 2
0
2
0 1 The Toeplitz lattice and its Virasoro algebra 12
/
h 1.1 The Toeplitz lattice . . . . . . . . . . . . . . . . . . . . . . . . 12
p 1.2 Virasoro constraints . . . . . . . . . . . . . . . . . . . . . . . . 26
-
h
t 2 Rational recursion relations 31
a
m 2.1 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
: 2.2 Rational recursion relations . . . . . . . . . . . . . . . . . . . 33
v
i 2.3 Proof of main Theorem . . . . . . . . . . . . . . . . . . . . . . 39
X
r
a 3 Invariant manifolds for the first Toeplitz flow 41
∗Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. E-
mail: [email protected]. The support of a National Science Foundation grant #
DMS-01-00782is gratefully acknowledged.
†Department of Mathematics, Universit´e de Louvain, 1348 Louvain-la-Neuve, Bel-
gium and Brandeis University, Waltham, Mass 02454, USA. E-mail: vanmoer-
[email protected] and @math.brandeis.edu. The support of a National Science Foun-
dation grant # DMS-01-00782, a Nato, a FNRS and a Francqui Foundation grant is
gratefully acknowledged.
1
4 Rational relations for special weights 43
4.1 Weight et(z+z−1) . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Weight et(z+z−1)+s(z2+z−2) . . . . . . . . . . . . . . . . . . . . . 44
4.3 Weight (1+z)αe−sz−1 . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Weight (1 ξz)α(1 ξz−1)β . . . . . . . . . . . . . . . . . . . 47
− −
5 Appendix 1: Virasoro algebras 48
6 Appendix 2: Useful formulae about the
Toeplitz lattice 49
7 Appendix 3: Proof of Theorem 0.2 50
In a discussion in spring 2001, Alexei Borodin showed us recursion re-
lations for the Toeplitz determinants going with the symbols et(z+z−1) and
(1 ξz)α(1 ξz−1)β. BorodinobtainedtheserelationsusingRiemann-Hilbert
− −
methods; see the recent work of Borodin [5] and also Borodin-Deift [6] and
Baik [4]. The nature of Borodin’s recursion relations pointed towards the
Toeplitz lattice and its Virasoro algebra, introduced by us in [3]. In this
paper, we take the Toeplitz and Virasoro approach for a fairly large class of
symbols, leading to a systematic way of generating recursion relations. The
latter are very naturally expressed in terms of the L-matrices appearing in
the Toeplitz lattice equations. As a surprise, we find, compared to Borodin’s,
a different set of relations, except for the 3-step relations associated with the
symbol et(z+z−1).
0 Introduction and main results
The weight
ρ(z) := eP1(z)+P2(z−1)zγ(1 d z)γ1′(1 d z)γ2′(1 d−1z−1)γ1′′(1 d−1z−1)γ2′′
− 1 − 2 − 1 − 2
(0.0.1)
with
N1 u zi N2 u zi
i −i
P (z) := and P (z) := , (0.0.2)
1 2
i i
1 1
X X
2
has a natural involution
˜: z z−1, (0.0.3)
↔
which induces an involution on the following quantities:
˜ : P (z) P (z−1), u u , N N , γ γ,d d−1, γ′ γ′′.
1 ↔ 2 i ↔ −i 1 ↔ 2 ↔ − i ↔ i i ↔ i
(0.0.4)
The multiple integral below is known to be expressible, both, as the deter-
minant of a Toeplitz matrix and as an integral over the group U(n),
n
1 dz
I(ε) := ∆ (z) 2 zερ(z ) k
n n! | n | k k 2πiz
Z(S1)n k=1(cid:18) k(cid:19)
Y
dz
= det zε+i−jρ(z)
2πiz
(cid:18)ZS1 (cid:19)1≤i,j≤n
= det(Uερ(U))dU, (0.0.5)
ZU(n)
which for some special choices of ρ has an interesting interpretation in terms
of random permutations; for that matter, look at the examples in section 4.
Consider the basic variables, with I := I(0), I± = I(±1),
n n n n
I+ I− I I
x = ( 1)n n , y = ( 1)n n and v = 1 x y = n−1 n+1. (0.0.6)
n − I n − I n − n n I2
n n n
See section 1.1 for explanations. Then the basic object
dz
I = det(ρ(U))dU = det zi−jρ(z) ,
n
2πiz
ZU(n) (cid:18)ZS1 (cid:19)1≤i,j≤n
which appears in several problems of random words and permutations, is
obtained from the x , y and I , by means of the formula
n n 1
n−1
I = In (1 x y )n−i. (0.0.7)
n 1 − i i
1
Y
3
Thefollowingmatrices, intimatelyrelatedtotheToeplitzlattice, willplay
an important role in this work :
x y 1 x y 0 0
1 0 1 1
− −
x y x y 1 x y 0
− 2 0 − 2 1 − 2 2
L1 := −x3y0 −x3y1 −x3y2 1−x3y3 (0.0.8)
x y x y x y x y
− 4 0 − 4 1 − 4 2 − 4 3
...
and
x y x y x y x y
0 1 0 2 0 3 0 4
− − − −
1 x y x y x y x y
− 1 1 − 1 2 − 1 3 − 1 4
L2 := 0 1−x2y2 −x2y3 −x2y4 . (0.0.9)
0 0 1 x y x y
− 3 3 − 3 4
...
The Toeplitz lattice and its relation to the Toda lattice will be discussed in
Section 1.1.
Define the matrices, depending on the positive integer n 1, and the
≥
exponents γ,γ′ and γ′′ in (0.0.1),
i i
(n) := (aI +bL +cL2)P′(L )+c(n+γ′ +γ′ +γ)L
L1 1 1 1 1 1 2 1
(n) := (cI +bL +aL2)P′(L )+a(n+γ′′ +γ′′ γ)L , (0.0.10)
L2 2 2 2 2 1 2 − 2
and depending on arbitrary parameters a,b,c. The involution˜, defined in
(0.0.3) and (0.0.4) induces involutions
I I , I+ I−,x y , a c, b b, and so L L⊤, (n) (n)⊤.
n ↔ n n ↔ n n ↔ n ↔ ↔ 1 ↔ 2 L1 ↔ L2
(0.0.11)
Also note that (self-dual case)
ρ(z) = ρ(z−1) implies x = y , L = L⊤, (n) = (n)⊤. (0.0.12)
n n 1 2 L1 L2
Given a matrix A(n) containing explicitly the parameter n, the “discrete
derivative” ∂ is defined as
n
∂ A(n) := A(n+1) A(n) . (0.0.13)
n nn n+1,n+1 nn
−
4
Rational relations: In the Theorem 0.1, we show the polynomial relation-
ships between consecutive (x ,y )’s. When the degrees of P and P differ by
i i 1 2
at most one, they actually lead to inductive rational relations, as is stated
in Theorem 0.2. These relations are obtained by observing that the multiple
integral (0.0.5) satisfy the Toeplitz lattice and an SL(2,Z)-set of Virasoro
relations in the u -variables; see [3].
i
Theorem 0.1 For the weight (0.0.1), the vectors (x ) and (y ) satisfy
k k≥1 k k≥1
two finite difference relations, involving a finite number of steps:
Case 1. When d ,d ,d d , γ′ + γ′′ , γ′ + γ′′ = 0 in the weight
• 1 2 1 − 2 | 1| | 1| | 2| | 2| 6
(0.0.1), then the relations are
(n) (n)
∂ ( ) + (cL aL ) = 0 (0.0.14)
n L1 −L2 n,n 1 − 2 nn
∂ (v (n) (n)) + cL2 +bL C = 0, (0.0.15)
n nL1 −L2 n+1,n 1 1 n+1,n+1 −
(cid:0) (cid:1)
for all n 1, and where
≥
a = 1, b = d d , c = d d .
1 2 1 2
− −
C is a constant independent of n, thus expressable in terms of the initial
value:
C := (v (1) (1)) + cL2 +bL . (0.0.16)
1L1 −L2 2,1 1 1 1,1
(cid:0) (cid:1)
Only the first relation is self-dual for the involution˜.
Case 2. When d = 0,γ′ = 0, γ′ = γ′′ = γ′′ = d = 0, we may rescale z
• 1 6 1 6 2 1 2 2
so that d = 1 and so
1
−
ρ(z) = zγ(1+z)γ1′eP1(z)+P2(z−1).
Then the same equations (0.0.14) and (0.0.15) are satisfied, where a,b,c can
be chosen in two different ways, one being the dual of the other, namely
(a,b,c) = (1,1,0) or (a,b,c) = (0,1,1).
Case 3. d = d = γ′ = γ′ = γ′′ = γ′′ = 0. Then
• 1 2 1 2 1 2
ρ(z) := zγeP1(z)+P2(z−1)
5
and a,b,c can be chosen totally arbitrary. Here it will be more advantageous
to pick different relations, both of which are polynomials, dual to each other,
(L P′(L )) (L P′(L ))
v − 1 1 1 n+1,n+1 − 2 2 2 n,n
n
Γ (x,y) := +nx = 0
n n
y
n +(P′(L )) +(P′(L ))
1 1 n+1,n 2 2 n,n+1
(L P′(L )) (L P′(L ))
v − 1 1 1 n,n − 2 2 2 n+1,n+1
Γ˜ (x,y) := n +ny = 0
n n
xn +(P′(L )) +(P′(L ))
1 1 n+1,n 2 2 n,n+1
(0.0.17)
Theorem 0.2 Requiring N = N or N 1 in the weight (0.0.1), the x
1 2 2 n
±
and y ’s can be expressed rationally in terms of lower x’s and y’s, and thus,
n
from (0.0.7), I can be expressed in the terms of the x’s and y’s. To be
n
precise,
Case 1 leads to two inductive rational N +N +4-step relations
1 2
•
x = F (x ,y ,... ,x ,y )
n n n−1 n−1 n−N1−N2−3 n−N1−N2−3
y = G (x ,y ,... ,x ,y ).
n n n−1 n−1 n−N1−N2−3 n−N1−N2−3
Case 2 leads to two inductive rational N +N +3-step relations (0.0.14)
1 2
•
and (0.0.15), such that1
when N = N or N = N +1, use (a,b,c) = (1,1,0)
1 2 1 2
when N = N or N = N +1, use (a,b,c) = (0,1,1).
2 1 2 1
Thus, we find rational functions F and G :
n n
x = F (x ,y ,... ,x ,y )
n n n−1 n−1 n−N1−N2−2 n−N1−N2−2
y = G (x ,y ,... ,x ,y ).
n n n−1 n−1 n−N1−N2−2 n−N1−N2−2
Case 3 leads to two inductive N +N +1-step rational relations
1 2
•
x = F (x ,y ,... ,x ,y )
n n n−1 n−1 n−N1−N2 n−N1−N2
y = G (x ,y ,... ,x ,y ).
n n n−1 n−1 n−N1−N2 n−N1−N2
1Both solutions can be used, when N =N .
1 2
6
Corollary 0.3 For the self-dual weight
ρ(z) = e N1 uii(zi+z−i),
P
the polynomial2 in x ,x ,... ,x , ... ,x ,
k−N k−N+1 k k+N
N N N
v
Γ := kx k u Li + u Li 2 u Li−1
k k− xk i 1! i 1! − i 1 !
X1 k+1,k+1 X1 k,k X1 k+1,k
N k+i−1 k+1
= kx +v u x (Li−1) + x (Li−1) =0
k k i j+1 1 k+1,j+1 j−1 1 jk
1 j=k−1 j=k−i+1
X X X
(0.0.18)
leads to recurrence relations
x = F (x ,... ,x ).
n n n−1 n−2N
Remark: In the self-dual case, i.e., when ρ(z) = ρ(z−1), the first equation
(0.0.14) vanishes identically and the two equations in (0.0.17) become iden-
tical. Only one equation is required, since all x = y .
n n
Remark: By the duality a c, b b, L L⊤, equations (0.0.14) and
↔ ↔ 1 ↔ 2
(0.0.15) map into
(n) (n)
∂ ( ) + (cL aL ) = 0
n L1 −L2 n,n 1 − 2 nn
(0.0.19)
∂ ( (n) v (n)) aL2 +bL = C′,
n L1 − nL2 n,n+1 − 2 2 n+1,n+1
where C′ is now: (cid:0) (cid:1)
C′ := ( (1) v (1)) aL2 +bL . (0.0.20)
L1 − 1L2 1,2 − 2 2 1,1
An invariant manifold: The relations ap(cid:0)pearing in(cid:1)Theorem 0.1 for each
of the cases happen to define an invariant manifold for the first Toeplitz flow.
We shall do this here for case 3, where
ρ(z) := zγeP1(z)+P2(z−1).
This case has the extra feature that the relations themselves (0.0.17) satisfy
an interesting system of differential equations. These statements will be es-
tablished in section 3, as an immediate consequence of the Virasoro relations
satisfied by the multiple integrals.
2The matrix L appearing in (0.0.18) is the matrix (0.0.8), with y =x .
1 i i
7
Theorem 0.4 Let x = x (t ,s ) and y = y (t ,s ) flow according to the
n n 1 1 n n 1 1
differential equations (v := 1 x y )
n n n
−
∂x ∂y
k k
= v x = v y
k k+1 k k−1
∂t ∂t −
1 1
(Toeplitz Lattice) (0.0.21)
∂x ∂y
k k
= v x = v y .
k k−1 k k+1
∂s ∂s −
1 1
and the u , appearing in the polynomials P (z) and P (z), according to
i 1 2
∂u ∂u
k k
= δ = δ .
k,1 k,−1
∂t ∂s −
1 1
Then:
(i) The polynomialrecurrence relationsΓ and Γ˜ , definedin (0.0.17), satisfy
n n
the differential equations
∂
Γ = v Γ +x x Γ˜ y Γ
n n n±1 n±1 n n n n
∂ t1 −
s1 (cid:16) (cid:17)
n o
∂
Γ˜ = v Γ˜ +y x Γ˜ y Γ . (0.0.22)
n n n∓1 n∓1 n n n n
∂ t1 − −
s1 (cid:16) (cid:17)
n o
(ii) The locus M is an invariant manifold for the t and s -flows (0.0.21)
1 1
above, where
M := (x ,y ) , such that Γ (x,y) = 0 and Γ˜ (x,y) = 0 .
k k k≥0 n n
n\≥0n o
(0.0.23)
Corollary 0.5 Let x = x (t) flow according to the differential equations
n n
(v := 1 x2)
n − n
∂x
n
= v (x x ), (0.0.24)
n n+1 n−1
∂t −
8
which is obtained by taking the linear combination ∂ ∂ of the Toeplitz
∂t1 − ∂s1
vector fields above and setting all x = y . Let the u , appearing in the
k k i
self-dual weight
ρ(z) = e N1 uii(zi+z−i),
P
flow according to
∂u
k
= δ .
k,1
∂t
Then:
(i) The polynomial recurrence relations Γ , as in (0.0.18), satisfy the dif-
n
ferential equations
∂Γ
n
= v (Γ Γ ). (0.0.25)
n n+1 n−1
∂t −
(ii) The locus N is an invariant manifold for the t-flow (0.0.24) above, where
N := (x ) , such that Γ (x,x) = 0 . (0.0.26)
k k≥0 n
{ }
n≥0
\
Singularity confinement: For the self-dual weight
ρ(z) = e N1 uii(zi+z−i),
P
the polynomial relations (remember Corollary 0.3) in x ,x ,... ,x ,
k−N k−N+1 k
... ,x ,
k+N
N N N
v
0 = kx k u Li + u Li 2 u Li−1
k− xk i 1! i 1! − i 1 !
X1 k+1,k+1 X1 k,k X1 k+1,k
lead, in effect, to rational recurrence relations in the x ,
i
x = F (x ,... ,x ;u ,... ,u ), (0.0.27)
k k k−1 k−2N 1 N
depending rationally on the coefficients u ,... ,u appearing in the weight
1 N
ρ(z).
The following Theorem tells us -roughly speaking- that the recurrence
relations (0.0.27) for special initial condition leads to a solution, where one
x blows up and all other x are finite. This is a kind of discrete Painlev´e
n k
property, called “singularity confinement”; see Grammaticos, Nijhoff and Ra-
mani[9], who definethistobediscrete Painlev´e recursion relations. Forthese
9
recurrence equations (0.0.27), the precise analytical statement of this phe-
nomenon is stated in Corollary 0.7, claiming there is a generic solution with
the kind of singularity above. The technique used here to prove Corollary
0.7 is to deform the variables x and y by means of the Toeplitz lattice;
k k
part (i) of Theorem 0.6 below shows that the Toeplitz lattice has a generic
solution x ,x ,..., with all x , k = n finite and one x blowing up. This
0 1 k n
6
is reminiscent of the Painlev´e property of algebraic integrable systems, which
originates in the work of S. Kowalewski; see [1] and references within. Part
(ii) of Theorem 0.6 shows that these series can be made to stay within the
locus N, by restricting the free parameters. The proof of Theorem 0.6 and
Corollary 0.7, which will be given in a subsequent paper, uses heavily the
ideas of Theorem 0.4 and Corollary 0,5.
Theorem 0.6 (i) Consider the system of differential equations (with bound-
ary condition x = 1)
0
∂x
k = (1 x2)(x x ), for k = 0,1,2,... , (0.0.28)
∂t − k k+1 − k−1
and for a fixed, but arbitrary integer n > 0, let
... ,α ,α ,c,d,α ,α ,... (0.0.29)
n−3 n−2 n+2 n+3
be free parameters. Then the system (0.0.28) has a unique “formal” Laurent
solution, with x and only x blowing up, having the form:
n n
x (t) = α +... , for k n 2
k k
| − | ≥
x (t) = 1+ct+...
n−1
±
(0.0.30)
x (t) = 1( 1 + c−dt+...)
n t ∓2 8
x (t) = 1+dt+... .
n+1
∓
The coefficients in the series (0.0.30) are polynomials in the free parameters
(0.0.29). This solution is generic, since
# free parameters +1 = # variables ,
{ } { }
with the “1” accounting for the t-parameter.
10
