AU0121208
Application of the r-function theory of Painleve equations to random matrices:
Pv , Pin , the LUE, JUE and CUE
P.J. Forrester and N.S. Witte* Department of Mathematics and Statistics ^(and School of Physics), University of Melbourne, Victoria 3010, Australia ; email: [email protected]; [email protected] With (•) denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is £W(/;a,£i) := (IIili X(o oo)\/(^ ~ A;)M) f°r I = (0>s) and I = (s;°°)i where xp = 1 for A; € / and xj — 0 otherwise. Using Okamoto's development of the theory of the Painleve V equation, it is shown that EN{I;CL,[I,) is a r-function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second order second degree differential equation. The cases fj, = 0 and \i = 2 are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case / = (s,oo), EN{I\CL,H) is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of £V(/;a,^). In particular, in the hard edge scaled limit it is shown that the limiting quantity £hard((0, s);a,fi) can be evaluated as a r-function associated with the Hamiltonian in Okamoto's theory of the Painleve III equation. keywords: random matrices - Painleve equations - Toda lattice equation - Backlund transformations - root systems - affine Weyl groups - classical orthogonal polynomial systems
1 Introduction and summary
In a previous paper [17] the quantities
and (n) (1.2) where the averages are with respect to the joint eigenvalue distribution of the Gaussian unitary ensemble (GUE), were shown to be equal to the r-functions occurring in Okamoto's theory [28] of the Painleve IV equation. It was noted in [17] that we expect the analogous quantities for the Laguerre unitary ensemble (LUE) to be expressible in terms of the -r-functions occurring in Okamoto's theory [29] of the Painleve V equation. It is the purpose of this article to verify this statement by giving the details of the correspondences between the multi-dimensional integrals defining the analogues of JSAT(A;O) and FN(\;O) for the LUE, and the r-functions from [29]. Let us first recall the definition of the LUE. Let X be a n x N (n > N) Gaussian random matrix of complex elements Zjk, with each element independent and distributed according to the Gaussian density ^e"'2-**' so that the joint density of X is proportional to
e-Mtx. (1.3)
Denote by A the non-negative matrix X'X. Because (1.3) is unchanged by the replacement X i-> UXV for U a n x n unitary matrix and V a JV x JV unitary matrix, the ensemble of matrices A is said to have a unitary
32/ 40 symmetry. The probability density function (PDF) for the eigenvalues of A is given by
1 N c n 1=1 where C denotes the normalization and a = n — JV, n > N. (Throughout, unless otherwise stated, the symbol C will be used to denote some constant i.e. a quantity independent of the primary variables of the equation.) Because Aae~* is the weight function occurring in the theory of the Laguerre orthogonal polynomials, and the ensemble of matrices A has the aforementioned unitary symmetry, the eigenvalue PDF (1.4) is said to define the LUE. Analogous to EN(X;O) specified by (1.1) for the GUE, we introduce the quantities
)LUE
EN ((a, oo); a, p) := ( f[f 5$ (s -A,)") (1.6) where the averages are with respect to (1.4) (the parameter a in (1.1) has been denoted /J, in (1.5), (1.6) to avoid confusion with the parameter a in (1.4)). Explicitly
EN((0,s);a,fi) 1 /*oo /"oo •*• / /-7\ \^rt — 1 (\ \iii I i-7\ \^ — N (\ "^^ T F (\ \ \^
^ \ f „ \ 1—r o /
TT (A* " \ In the case /i = 0 the first integrals in (1.7) and (1.8) are the definitions of the probability that there are no eigenvalues in the intervals (0, s) and (s, oo) respectively of the Laguerre unitary ensemble. The case p, — 2 also has significance in this context. To see this, first note from the definitions (using the first integral representation in each case) that a ^-EN+1 ((0, a); a, 0) oc s e—EN((0, s); a, 2) (1.9) a s •fEN+1{(s,oo);a,0) oc 8 e- EN((s,oo);a,2). (1.10) CIS On the other hand, with pmin(s; a) and pm&x(s; a) denoting the distribution of the smallest and largest eigenvalue respectively in the N x N LUE, we have pTnin(s;a) = —^EN{{O,s);a,O) (1.11) as Pmax(s;a) = -T- EN((s,oo);a,0). (1.12) Hence for JV i-> JV+1, pmin(s;o) andpmax(s; o) are determined by EN((0, S); a, 2) and J5AT((S,OO);O, 2) respectively. From the second formula in (1.7), we see that with N ) (1-13) 'LUE we have Ns FN(s;a,fJ,)= (e EN((O,s);n,a))\ (1.14) (notice the dual role played by fi and a on the different sides of (1.14)) so there is no need to consider FM separately. Note that with /j, = 2, (1.13) multiplied by sae~s is proportional to the definition of the density in the LUE with N *-+ N + 1. The second integral in (1.8) is of interest for its relevance to the Jacobi unitary ensemble, which is specified by the eigenvalue PDF N This ensemble is realized by matrices of the form A(A + B)"1, where A = X^X, B = Y^Y, for X (Y) annixJV (n2 x N) complex Gaussian random matrix with joint density (1.3). The parameters o and 6 are then specified by a = ni — N, b — ri2 — N (c.f. the value of o in (1.4)). We see from (1.15) that 1 A a b sX 2 s ~ f dAi Af (1 - AX)V > • • • / d\N \ N(l - \N) e » TT (Afc - A,) = (e ^=^ *') , (1.16) CJ° Jo i JUE where Xl a XN X IN[a):= [ d\i\1e- ••• [ d\N \ Ne~ ]J ( k - Xjf (118) 2 JN(a,fj,):= f dAiA;(l-Ai)""- / dA^A^a-Aivf J][ (\k-\j) . (1.19) Jo Jo l As is well known, the integrals IN (a) and Jjv(a,/u) can be evaluated in the form Af!n=o CI> where Cj is the normalization of the monic orthogonal polynomial of degree j associated with the weight functions Aae~A and Aa(l-A)'i respectively. One important feature of the JUE is that with the change of variables (cos^ + 1), (1.20) the PDF (1.15) assumes a trigonometric form, which for appropriate (TV, a, 6) coincides with the PDF for the independent eigenvalues e10' of random orthogonal and random unitary symplectic matrices. In the case of orthogonal matrices, one must distinguish the two classes 0% and O^ according to the determinant equalling +1 or —1 respectively. The cases of ./V even and N odd must also be distinguished. For N odd all but one eigenvalue come in complex conjugate pairs e±t9j, with the remaining eigenvalue equalling +1 for O% and —1 for OJ,. For N even, all eigenvalues of O~^ come in complex conjugate pairs, while for Ojf all but two eigenvalues come in complex conjugate pairs, with the remaining two equalling ±1. Let us replace N in (1.15) by N* and make the change of variables (1.20). Then the PDF for the independent eigenvalues of an ensemble of random orthogonal matrices is (see e.g. [10]) of the form (1.15) with (JV/2,-1/2,-1/2) for matrices in O£, iVeven NN 22 (N* b) = J dd ~~ i)/i)/ ' ' -1/2,1/2-1/2,1/2)) foforr matricematricess iinn O+,O+, AATT od oddd 1 'a' > > ((JV-l)/2,1/2,-1/2((JV-l)/2,1/2,-1/2)) foforr matricematrices iin O~O~,, AAT od oddd ; (JV/2 - 1,1/2,1/2) for matrices in O^, JVeven Matrices in the group USp(N) are equivalent to 2JV x 2N unitary matrices in which each 2x2 block has a real quaternion structure. The eigenvalues come in complex conjugate pairs e », and the PDF of the independent elements is of the form (1.15) with the change of variables (1.20) and {N\a,b) = {N,1/2,1/2). (1.22) It follows from the above revision that (esT«u)) (1.23) for G = O%,O^ or USp(N) is a special case of the more general average (1.16). Explicitly, T y / < >\ ""/2^x\ (124) uea \ /JUE where on the RHS the dimension of the JUE is at first N*, then the parameters (JV*, a, b) are specified as in (1.21) or (1.22). Also, \N- = 0 for G = USp(N) and O%,O^ (N even), while XN- = ±1 for G = O%,Ojj (N odd) respectively. The averages (1.23) have independent importance due to their occurrence as generating functions for certain combinatorial problems. Let us quote one example. Set /„; equal to the number of fixed point free involutions of {1, 2,..., 2n} constrained so that the length of the maximum decreasing subsequence is less than or equal to 21, and introduce the generating function According to a result of Rains [31] (see also [5]) we have mm) (1.26) ueusP(i) A list of similar results is summarized in [2]. The second integral in (1.8) can be written in a trigonometric form distinct from that which results from the substitution (1.20). For this purpose we recall the integral identity [13] fl e-i f1 , e-i I dtiti ••• I dtwtN f(ti,... ,tNj Jo Jo , „ AT /"1/2 z-1/2 x 2nieX1 2 ieXN 2jrieXl 2 ieXN = {-A—) / dxie --- dxNe * f(-e ,...,-e " ) (1.27) VsmTre/ J-1/2 J-1/2 valid for / a Laurent polynomial. Applying (1.27) to (1.19) gives the trigonometric integral M (a',b'):= f dx e*i*l{a'-b')\\ + e2"ixi\a'+b' ••• f dx 'iXN(a'-b>)\ N rl/2 l N e J-l/2 J-l/2 -1/2 J-1/2 x TT \e27riXk -e2™*'!2. (1.28) Moreover, applying (1.27) to the LHS of (1.16) and using Carlson's theorem it follows that ^^'"') 2^ f^'^" ) (1.29) r_ /CUE where a' = N + a + fi, b' — —(N + a) and the CUE (circular unitary ensemble) average is over the eigenvalue PDF ix 2 x 2 I Yl \e^ "-e " '\ , -1/2 < xi < 1/2. (1.30) l Recalling (1.17) then gives 6 |1+e 6 S l ' /cuE~M^(0,0)Jw(a,M) *.NU8,ca),attt), (1.31) although the RHS is convergent for only a subset of the parameter values for which the LHS converges (explicitly the RHS requires Re(o' + 6') > —1 and Re(—N — b') > —1 to be properly defined, while the LHS only requires Re(a' + b') > —1). Note however that by replacing the integrals over A; £ [0,1] in (1.8) by the Barnes double loop integral [40] the quantity EN((S,OO);O,,II) has meaning for general values of a and fi. In the case a' = 0, b' = ke Z>0, the LHS of (1.31) reads (( )) (1.32) CUE \ /uecvE Like (1.26) this latter average with the dimension N replaced by / is the generating function for certain combi- natorial objects [37] (random words from an alphabet of k letters with maximum increasing subsequence length constrained to be less than or equal to I). As we have already stated, our interest is in the evaluation of (1.8) and (1.7) as r-functions occurring in Okamoto's theory of Pv • In some cases such evaluations, or closely related evaluations, are already available in the literature. One such case is (1.8) and (1.7) with p, = 0, for which Tracy and Widom [34] (for subsequent derivations see [1, 20, 7]) have shown EN((0,s);a,Q)=exp f^-dt (1.33) Jo T dt L34 EN((s,oo);a,0) = exp(- H ^ ) ( ) where a(t) satisfies the Jimbo-Miwa-Okamoto cr-form of the Painleve V equation, 2 2 (to") - [a - to' + 1(o'f + Wo + vi + V2 + v3)o'] + 4(i/0 + a')Wi + a')(y2 + a')(v3 + a) = 0, (1.35) with i/0 = 0, i/i=0, i/2 = N + a, u3 = N (1.36) (or any permutation thereof, since (1.35) is symmetrical in {fk}), and subject to appropriate boundary conditions in each case. For general fi the second integral in (1.8) has been written in an analogous form to the right hand side of (1.33) by Adler and van Moerbeke [2], with the function corresponding to a presented as the solution of a certain third order differential equation. It is remarked in [2] that this equation can be reduced to a second order second degree equation, known to be equivalent to (1.35), using results due to Cosgrove [9, 8] (see also the discussion in the final section of [41]). The average (1.32) can be computed in terms of Painleve transcendents by making use of an identity [11] implying k sTl(u) ks (det(l + U) e- )ueCVE = e Ek((0, a);N, 0), (1.37) (note the dual role played by k and JV on the different sides of (1.37)) then appealing to (1.33) (direct evaluations have been given by Tracy and Widom [37] and Adler and van Moerbeke [2]). The Okamoto T-function theory of Pv provides a unification and extension of these results. Consider first (1.7). Proposition 6 below gives eXP dt IN (a) /0 t Na-Ns PY IN (a) where Vjv(£;a, /J) satisfies the Jimbo-Miwa-Okamoto a form of Pv (1.35) with VQ = 0, vi = -fi, v2=N + a, i>3 = N, (1.39) and subject to the boundary condition N(N >Ai) ~ -Nt + N(a - p) - * ^ + O(4)- (1-40) Regarding (1.8), in Proposition 5 below we show exp ^4IN (a) where like Vjv(i;o,/a), £//v(£;a,/i) satisfies (1.35) with parameters (1.39), but with the boundary condition (1.42) By substituting the first equation of (1.41) in (1.17) and (1.31) we obtain the evaluations = exp /JUE 6i-»/i CUE M,v(a',&') >- (L44) Comparison of (1.43) with (1.24) gives that for G = ON,USp(N), exp with (N*,a,b) specified by (1.21) or (1.22) as appropriate. In particular, with G — USp(N) and thus (N*,a,b) given by (1.22), substitution into (1.26) shows Pl(s) = exp * After recalling the statement below (1.12) and inserting the proportionality constants in (1.9) and (1.10), the evaluations (1.38) and (1.41) give Pmin(s;a)i ^ =(N<™ + . 1)——1^0 —+ r^T2) sea -s exP /f (1.47) N^-iN+l lN+l{a) Jo _ f° UN{t;a,2) Pmax(s;a) s o+2Ar e s ex dt ). (1.48) P Js t J Also, recalling (1.14) and the sentence below that equation, we see from (1.38) that p{s) /_ f° \ J_s t where p(s) denotes the eigenvalue density in the LUE. A generalization of the integral identity (1.37) is derived in the context of the Painleve theory in Proposition 10 below. We remark that this generalization is in fact a special case of a still more general identity, known from an earlier study [11], involving an arbitrary parameter /3. For f3 = 1 an average in the Laguerre orthogonal ensemble is related to an average in the circular symplectic ensemble, while for j3 = 4 an average in the Laguerre symplectic ensemble is related to an average in the circular orthogonal ensemble. The Laguerre ensemble permits four scaled, large N limits. These are [12, 6, 23, 24] s H- s/4iV, N ->• oo (1.50) s i->- 4iV + 2(2iV)1/3s, AT ->• oo (1.51) 2 a = (7 - 1)JV, s>->N(l- ^/y) - v- (N)s, N -> oo, (1.52) o = (7 - 1)N, s^>N(l + V7)2 + v+(N)s, N ->• 00, (1.53) 6 where in (1.52), (1.53) we require 7 > 1 and 1/3 / 1 \ 1/3 (N){N(f + l){l+ +)) (1-54) The first gives the limiting distributions at the hard edge, so called because the eigenvalue density is strictly zero on one side. The second, third and fourth give the limiting distribution at the soft edge, so called because there is a non-zero density for all values of the new coordinates, but with a fast decrease on one side. The soft edge limit for the quantities (1.1) and (1.2) in the GUE was studied in our work [17], where evaluations in terms of solutions of the general Jimbo-Miwa-Okamoto cr-form of Pn (u"f + Au {[uf - tu + u) - a2 = 0. (1.55) were given. Prom either of the limiting procedures (1.51), (1-52) or (1.53) we reclaim the results of [17] (because no new functional forms appear we will not discuss this case further in subsequent sections). In particular, it follows from the second equation in (1.41) that Eso{t{s;fi)= lim Ce-W2£((s, = lim ^~r—(uN(t;a,fj,) + Nfi-^t)\ . (1.57) N->oo AN \ 2 / lt>4JV+2(2A01/St Making the replacements %t, t^AN + 2(2N)l/3t, a(AN + 2(2iV)1/3i) in (1.35) with parameters (1-39), and equating terms of order N2 in the equation (which is the leading order) shows u(t;fi) satisfies (1.55) with a = \i. We know from [17] that (1.55) is to be solved subject to the boundary condition As an application, (1.57) can be used to compute the scaled limit P(s):= lim 1+00 Imposing the condition that P{s) —$• 1 as 8 —¥ 00 (which fixes the constants), a short calculation using (1.57) in (1.46) shows P(s) = exp ( - I (u(-t; 1/2) - t2/A) di). (1.59) This is interesting because it has been proved by Baik and Rains [4] that 1/3 P(2 s) = F1(s), (1.60) where F\(s) is the cumulative distribution function for the largest eigenvalue in the scaled, infinite Gaussian orthogonal ensemble. Tracy and Widom [36] (see [15] for a simplified derivation) have shown that a Fi(a) = e-U~<«->« W<"eU~ «<*><" (1.61) where with Ai(i) denoting the Airy function, q{t) is the solution of the non-linear equation q"=2qs+tq, (1.62) (Painleve II equation with a = 0) subject to the boundary condition q(t) ~ -Ai(t) as t -»00. (1.63) By equating (1.59) and (1.60) we obtain as an alternative to (1.61) the formula 1/3 2 JF1(2- s) = expf- / (u(-t;l/2)-t /4)dt). (1.64) J s However the boundary condition (1.58) gives u(—t; 1/2) — £2/4 ~ 0 which is vacuous, whereas the boundary condition according to the requirement (1.63) is 1/3 2 2~ ;£(«(-*; 1/2)-* /4) 1/s ~ The significance of (1.64) from the viewpoint of gap probabilities in random matrix ensembles as r-functions for Hamiltonians associated with Painleve systems will be discussed in a separate publication [19]. For the hard edge scaling (1.50), define the scaled version of (1.7) by ) (1.65) In the case n = 0, it is known from the work of Tracy and Widom [35] that £hard(s; a, 0) = exp ( - j" ^&- dt) (1.66) where as(t) satisfies the Jimbo-Miwa-Okamoto a-form of the Painleve III equation 2 2 (ta") - wi«2( v\ = a, V2 = a. (1.68) It follows from Proposition 16 below that for general fi E^(s;a,,) = exp ( - [ ^ + ^ + ^ dt) (1.69) where a satisfies (1.67) with vi=a + fx, V2 = a —/J, (1-70) and subject to the boundary condition One application of (1.69) is to the evaluation of ^(^) (1.72) First we note that the explicit evaluation of (1.18) is YljLi r(l +j)T(a + j), which together with the asymptotic formula T(x + a)/T(x) ~ xa for x —> oo shows that lim —(„ -*,-„ , , „ „ ~ —'""(a + l)r(a + 2)' Using this formula and (1.69) we can take the large N limit in (1.47) as required by (1.72) to conclude where a(t) satisfies (1.67) with vi = a + 2, V2 = a — 2 and is subject to the boundary condition (1.71) with \i = 2. Similarly, it follows that the hard edge density is given by hard.x sae~s/4 ( f° g(t)+a(q + 2)/2^ , P {s)=2^r(a + l)r(a + 2)eXp{J 1 V (L74) where a(t) satisfies (1.67) with vi = 2 + a, V2 = 2 — a and is subject to the boundary condition (1.71) with (a,fj.) = (2,o). The result (1.69) also has relevance to the GUE. This occurs via the scaled limit of (1.37), which has been shown [11] to imply for o € Z>o, p € Z, the identity £hard(i;a,p) oc e-'/V^/ei^^+^detI/)"") (1.75) (eq. (4.32) below; the proportionality constant is specified in (4.33)). It thus follows that this CUEO average can be evaluated in terms of the transcendent a(t) in (1.69) (in the case fi = 0 this has been shown directly in the previous works [38, 2]). This concludes the summary of our results. In the next section (Section 2) an overview of the Okamoto r-function theory of the Painleve V equation is given. In particular a sequence of r-functions r[n], each of which can be characterized as the solution of a certain second order second degree equation, is shown to satisfy a Toda lattice equation. The fact that a special choice of initial parameters in the sequence permits the evaluations r[0] = 1 and rfl] an explicit function of t (a confluent hypergeometric function) then implies that the general member of the sequence r[n] is given explicitly as a Wronskian type determinant depending on r[l]. In Section 3 the Wronskian type determinants from Section 2 are evaluated in terms of the multiple integrals (1.7) and (1.8), thus identifying them as r-functions and implying their characterization as the solution of a non-linear second order second degree equation. In Section 4 the Okamoto r-function theory of the Painleve III equation is revised, and again a r-function sequence is identified which can be shown to coincide with a quantity in the LUE (explicitly £;hard(s;o,^i) for general fj, and a € Z>o). By scaling results from Section 3 it is shown that for general \x and general a, l?hard(s; a, fj.) as specified in (1.65) is a r-function in the Pm theory. Concluding remarks relating to the boundary conditions (1.71) and (1.58) are given in Section 5. 2 Overview of the Okamoto r-function theory of Py 2.1 The Jimbo-Miwa-Okamoto a-form of Py As formulated in [29], the Okamoto r-function theory of the fifth Painleve equation Pv is based on the Hamiltonian system {Q, P, t, K} 2 2 2 tK = Q(Q - 1) P - {(«2 - vi)(Q - I) - 2(wi + v2)Q{Q - 1) + tQ}p + («s - «i)(t»4 - t>i)(Q - 1) (2.1) where the parameters vi,..., VA are constrained by vi + v2 + v3 + v 4 = 0. (2.2) The relationship of (2.1) to Pv can be seen by eliminating P in the Hamilton equations Q ~ dP' F~ dQ- [2-6> One finds that Q satisfies the equation with s 2 5 a = g ("3 - Vif, p = --(V2-v1f, 7 = 2«i + 2t)2 - 1, =-^ ( - ) This is the general Pv equation with 5 — — | (recall that the general Pv equation with 5 ^ 0 can be reduced to the case with 5 = —\ by the mapping t M- \J—25t). Note that the first of the Hamilton equations (2.3) implies 2 tQ' = 2Q{Q - ifP - {{V2 - vi)(Q - I) - 2(«! + v2)Q(Q - 1) + tQ}, (2.6) which is linear in P, so using this equation to eliminate P in (2.1) shows that tK can be expressed as an explicit rational function of Q and Q'. Of fundamental importance in the random matrix application is the fact that tK (or more conveniently a variant of tK obtained by adding a term linear in t) satisfies a second order second degree equation. Proposition 1. [21, 29] With K specified by (2.1), define the auxiliary Hamiltonian h by h - tK + (v3 - vi)(i>4 - vi) - vit - 2v\. (2.7) The auxiliary Hamiltonian h satisfies the differential equation 4 {th"f -(h- th' + 2(ft')2) 2 + 4 ]J (h' + yk) = 0- (2.8) Proof. Following [29], we note from (2.1), (2.2) and the Hamilton equations (2.3) that h' = -QP - V! 2 2 2 th" = Q(Q - 1)P - |(«3 + Vi - 2vi)Q - vi + t>i}p + (v3 - vi)(t>4 - vi)Q. (2.9) We see from these equations that th" = Q(ti + V3)(ti + vi) + P(h! + v2), (2.10) while the first of these equations together with (2.1) and (2.7) shows 2 h-th' + 2(ti) = Q(h! + vs)(h' + vA) - P{h! + v2). (2.11) Squaring both sides of (2.10) and (2.11) and subtracting, and making further use of the first equation in (2.9) gives (2.8). • Actually the differential equation obtained by Jimbo and Miwa [21] is not precisely (2.8), but rather a variant obtained by writing (i) h = a -vjt-2v], .7 = 1,... ,4, or equivalently 00)) 22 <7 =tK+= (v3 - t)i)(v4 - «i) - («i - vj)t - 2(v - v]v]). (2.12) With this substitution (2.8) coincides with the Jimbo-Miwa-Okamoto a form of Pv (1.35) with {vO,Vi,V2,V3} = {Vl —Vj,V2—Vj,V3—Vj,Vi—Vj} (2.13) (because (1.35) is symmetrical in {fk} the ordering in the correspondence (2.13) is arbitrary). The r-function is defined in terms of the Hamiltonian by if = | log r. (2.14) It then follows from (2.12) that ( 1 ) 3 Ol)(V4 ol) 2( ? ?) 0.0) = t—log (e- " -"i V" - - - " -"> ry (2.15) dt \ 1 Now, according to the Okamoto theory the r-function corresponding to some particular sequences of parameter values can be calculated recursively in an explicit determinant form. The above theory tells us that these r- functions can also be characterized as the solutions of a non-linear differential equation. The determinant solutions are a consequence of a special invariance property of the Pv system which we will now summarize. 10 2.2 Backlund transformations A Backlund transformation of (2.1) is a mapping, in fact a birational canonical transformation where v,q,p,i,H are functions of v, q,p,t, H, such that the Pv coupled system (2.3) is satisfied in the variables (y;q,p,i). Formally dphdq- dH Adt = dp A dq- dH A di. In general, the significance of such a transformation is that it allows an infinite family of solutions of the Pv system to be obtained from one seed solution. Prom our perspective a key feature is the property T(H) = H (2.16) v>-»T-v which holds for some particular operators T which have a shift action on the parameter vector v = (vi,V2,vz,Vi) whose components are referred to the standard basis. Explicitly, following [29], we are interested in the transfor- mation To which has the property (2.16) with the action on the parameters . (2.17) Although the existence of To was established in [29] (termed the parallel transformation /(v)), as was the fact that To(t) = t, explicit formulas for To(P) and To(Q) were not presented. It is now realized [39, 25] that the required formulas are more readily forthcoming by developing the Pv theory starting from the symplectic coordinates and Hamiltonian {q,p, t, H} specified by tH := q(q - l)p(p + t) - (v2 - vi + v3 - Vi)qp + (v2 - vi)p + (vi - vz)tq. (2-18) The coordinates and Hamiltonians constituting the two charts (2.1) and (2.18) are related by the coordinate transformation [39] {q - l)p +(Q- 1)P = v3 - vi tK -tH + (i>3 - t>i)(t>2 - v4) (2.19) so in particular 1 — 1/g satisfies the Painleve V equation (2.4). The Hamiltonian (2.18) is associated with a set of four coupled first order equations in symmetric variables [3, 27] which admit an extended type A3 affine Weyl group Wa = Wa{A3^) » Z4 as Backlund transformations. The generators of the group Wa = {so, si,S2, S3, TT) obey the algebraic relations Si = 1, SiSi±iSi = Si±xSiSi±i, SiSj = SjSi (j ^ i,i± 1), 4 (2-20) TTSi = Si+lTT, 7T = 1, (i, j' = 0, . . . , 3, Si :== So). The parameters of Pv can also be characterized by the level sets of the root vectors ao, a\, 0.2, o-z (with 0.0 +<*i + a.2 + Q3 = S) of the A3 root system, and are dual to the vector parameter v in the following way In general the action of a group operation T on the roots is given in terms of its action on the components of v by T(a) • v = aCT1 • v) (2.21) Following [25] the action of the Backlund transformations on the root system a and coordinates q, p is given in Table 1. Note that with To defined by (2.17), TO(Q) = (ao — 1,01,02,0:3 + 1), 11 so according to Table 1 we have To = S3S2S17T, To = ITIT SS1S2S3. (2.22) As is the case for the dynamical system associated with Piv there is a symmetric form for the system describing the Pv transcendent [27], and is expressed in terms of the four variables p + t /o = Vt fi=>/iq (2.23) -q), with the constraints fo + J2 = fi + fa = \/i. Using these variables the Backlund transformations take the simple and generic forms of Si(ctj) = a>j - dijOn, 7T(QJ) = (2.24) SiUi) = fi ~ unfi, Afi) = fi+i, where the Cartan matrix A = (aij) and orientation matrix U = (tty) are defined by 2-1 0-1 0 10-1 -1 2-1 0 -10 10 A = (2.25) 0-1 2-1 0-1 0 1 -1 0-1 2 10-10 Let us now compute the action of To * on the Hamiltonian tK. First we can easily verify from Table 1 that so (tH) = tH + ao - + oto (a2 — 1) p + t = tH + ait + s2(tH) =tH- a2t + a2(ao - s3(tH) = tH + aia3 iritH) = tH+(q- l)p - a2t. (2.26) It follows from these formulas, Table 1 and (2.22) that 1 To (tH)=tH = tH + qp. (2.27) vi->Tov According to the relations in (2.18) between tH and tK, q,p and Q,P, we therefore have = tK - Q(Q - 1)P + {vi - vi)(Q - 1) = tK (2.28) which is the result deduced indirectly in [29]. Note from (2.19) that (2.29) where T^iq) and T^l(p) can be deduced from Table 1 using (2.22). 12 «O ai a2 as P so -a0 ai +a0 a2 as + ao P Si a0 + ai -ai a2 + ai a3 q s2 ao ai + a2 -a2 tt3 +O2 S3 ao + a3 "l a2 + 03 -a3 q IX ai Q2 a3 a0 t{q - 1) -f Table 1: Backlund transformations relevant for the Py Hamiltonian (2.18). 2.3 Toda lattice equation Introduce the sequence of Hamiltonians K[n] := K (2.30) and let r[n] denote the corresponding r-functions so that K[n] = — logr[n]. (2.31) We have already remarked that a crucial feature of K[n] from the viewpoint of application to random matrix theory is the fact that it (or more precisely the quantity (2.12)) satisfies the differential equation (1.35). An equally crucial feature for application to random matrix theory is the recurrence satisfied by {7"[n]}. Proposition 2. [29, 25] The T-junction sequence T[TI], corresponding to the parameter sequence (vi — n/4, V2 — n/4, «3 — n/4, «4 + 3n/4)7 obeys the Toda lattice equation 2 6 log T[TI] = (2.32) where (2.33) Proof. Following [25] we consider the sequence of Hamiltonians H[n] associated with the given parameter sequence and the Hamiltonian (2.18). Now, with f[n] denoting the corresponding r-function, it follows from (2.19) that Since the equation (2.32) is unchanged by the replacement ?[n] t-t ta+bnf[n], it suffices to show that (2.32) is satisfied with f[ra] replacing r[n] in (2.33). From the definitions f[ 1 1 + 1] Slog ""f a [^" == (ToHtH[n]) - tH[n]) - (tH[n] - T0(tH[n])). (2.34) On the other hand, it follows from (2.27) that 1 (To (tH[n\) - tH[n]) - (tH[n] - T0(tif [»])) = q[n]p[n] - To(«[n])T0(p[n]). (2.35) With T^1 specified from (2.22), explicit formulas for To(q[n}) and To(p[n]) can be deduced from Table 1, and it can be verified using the Hamilton equations for H that the RHS of (2.35) is equal to 5log fg[n](g[n] - l)p[n] + (vi - Vi)q[n] + (v4 - vi + n)J (o4 wi+n) = 5 log ~ (tH[n] + («4 -vi+ n)t) = <51og ^5 log (e - *f[n]). Equating this with the LHS of (2.34) gives a formula equivalent to (2.32). • 13 2.4 Classical solutions The Toda lattice equation (2.32) is a second order recurrence, and so requires the values of f[0] and f[1] for the sequence members f[n], (n > 2) to be specified. It was shown by Okamoto [29] that for special choices of the parameters, corresponding to the chamber walls in the underlying A3 root lattice, the Pv system admits a solution with r[0] = 1 and r[l] equal to a confluent hypergeometric function. Proposition 3. [29] For the special choice of parameters vi = V4 (2.36) it is possible to choose T[0] = 1. Furthermore, the first member r[l] of the T-function sequence (2.31) then satisfies the confluent hypergeometric equation t(T[l])" + (v3 - v2 + 1 + t)(r[l])' + (v3 - VI)T[1] = 0. (2.37) Proof. Write Q = Q[0], P = P[0]- In the case vi =• VA we see from (2.1) that it is possible to choose P = 0, K = 0, (2.38) the latter allowing us to take r[0] = 1. With (2.38) the equations (2.28), (2.1) and (2.31) then give tff IogT[l](t) = (t* - «i)(Q - 1). (2.39) The quantity Q must satisfy the Hamilton equation 2 l)-tQ (2.40) where use has been made of (2.2). Substituting (2.40) in (2.39) gives (2.37). • The two linearly independent solutions of (2.37) are a r [l](t) = iFi(t;3 - vi,v3 - v2 + 1; -t) 1 T{V3 ~ V2 + 1) f e-tuuvs-v1-lQ_uy1-v'•2 du (2.41) and rs[l](i) = e~trp(l + vi — V2, vz — V2 + 1; i) -t (the superscripts "a" and "s" denote analytic and singular respectively, and refer to the neighbourhood of t = 0). As noted by Okamoto [29], it is a classical result that the solution of the Toda lattice equation in the case f[0] = 1 is given in terms of r[l](<) via the determinant formula =o,...,n_i. (2.43) The formulas (2.41) and (2.42) substituted in (2.33) with V4 = vi and thus reading f[n] = tn2/2entr[n] (2.44) give the explicit form of f[l](i), and thus we have two distinct explicit determinant formulas for f[n]. 14 2.5 Schlesinger Transformations We are now in a position to derive difference equations for the dynamical quantities of the Pv system which are consequences of the Schlesinger transformations or shift operators corresponding to translations by the funda- mental weights of the Az lattice To = S3S2Sin, Ti=irs3S2Si, T2 = S1-KS3S2, TS = S2SITTS3. (2.45) It is well known that these difference equations can be identified as discrete Painleve equations [33] satisfying integrable criteria analogous to the continuous ones. Here we briefly demonstrate this and find a difference equation for the Hamiltonians. Proposition 4. [33] The Schlesinger transformation of the Pv system for the shift operator T^1 generating the parameter sequence (ao + n, a\, 012,03 — n) with n € Z, corresponds to the second order difference equation of the discrete Painleve type, <2Piv , namely t a3-n Vn 1 - Vn , 1 . • (2-46) where xn = fo[n]fi[n] — \a\ and yn = %/t//i[w]. With qp[n] — tH[n+l] —tH[n], the Hamiltonian times t, tH[n], satisfies the third order difference equation (ao+n+qp[n])[tH[n] + (03— n)qp[n] — (qp[n]-a3+n)[tH[n] + (l-«o-n)qp[n] + ait] + (qp[n-l]-ai)[tH[n] + (l-ao-n)qp[n] - (a2+as-n)t] tH[n] + (1—ao — n)qp[n] + (1—ao— n)(qp[n— 1]—QI) + ait (2.47) Proof. The Schlesinger transformations for the shift operator To that are relevant for the identification are . ao ao+ai+a2 To" *(/<>) = h-~r+ , (2.48) Jo + a0 (2.49) a0 (2.50) a0 /1 + 7- /o Using (2.50) in (2.48) along with the constraint to eliminate 73 we can rewrite the latter equation as ao+n _ n+1—03 This identity can then be employed in (2.49) to arrive at /0N/1N + /o[n-l]/i[n-l] = ai+ Vtfi[n] + "~az fi[n], (2.52) V t - /1 [n] 15 which is the first of the coupled equations (2.46). The second equation of the set (2.46) follows immediately from (2.49) rewritten to read 1 1 fo[n]fi[n]+ao+n /i[r»]/i[n+l] fo[n]fi[n] /o[n]/i[n]-ai ' l ' The simplest way to derive the difference equation for tH[n] is to recast the Schlesinger transformations in terms of canonical variables q,p, ao+ai+n tq[n+ l]=t+ p[n] , (2.54) a0 + n pln+1] = -tq[n] - "—-j-L + ^ -J-— , (2.55) -p[n] -\ ao+n q[n] + JTpM tq[n-l] = -p[n}+ + , (2.56) i — q\n\ a2+Q,3_n q[n] + as-n (a2+a3n)t t + p[n-l] = tq[n] + . (2.57) [] Considering the first two equations (2.54,2.55) we find that [ao+n+q[n](t+p[n])][qp[n]{t+p[n]) - aip[n] a2- p[n][ai-q[n}(t+p[n])] + (ao+a1+n)t and expressing this in terms of the Hamiltonian yields nnln + M + n, [tq[n]+ao+n+qp[n]][tH[n] + {as-n)qp[n] - a2t] [ + (a3-n-l)tq[n] + (a0+ai+n)f ' In an analogous way we find for the down-shifted product nr,lr,-i1-n \p[n]+a3-n-qp[n]][tH[n] + (l-ao-n)qp[n] + ait] m J J K ~ tH[n] + (l-ao-n)qp[n]-(l-ao-n)p[n]-(a2+a3-n)f ' Now the up-shifted equation can be solved for q and the down-shifted one for p allowing their product qp[n] to be expressed in terms of just the Hamiltonian and other members of the qp sequence. The final result is (2.47). • 3 Application to the finite LUE 3.1 The case N = 1 Comparing the case iV = 1 of the integrals (1.7) and (1.8) with those in (2.41) and (2.42) we see that Recalling (2.15), we thus have that . d 16 where both as<-^(t) and a^l\t) satisfy the Jimbo-Miwa-Okamoto a form of Pv (1.35) with vo = 0, v\ = — /J,, 1/2 = a + 1, ^3 = 1- Note that in the case p = 0 this is consistent with (1.36). 3.2 The general N case Although it is not at all immediately obvious, the n x n determinant formed by substituting (2.41) and (2.42) in (2.43) can be identified with the general n cases of the integrals (1.7) and (1.8) respectively. Consider first (2.43) with initial value (2.41). Proposition 5. Let f[n] be specified by the determinant formula (2.43) with 1/2 f[l]{t)=t e\F1(v3-vuV3-V2 + l;-t). (3.1) Then we have n2/2 tui {V2 vl V3 v n tu V2 vl) V3 v n 2 r[n]oct f duie u- - \\-ux) - i- --- [ dune "u^ - (l-un) - i- ]J (uk-Uj) . J° J° \ It follows from this that )N+N2 EN((t,oo);a,ii) = C&+» r[N](t) (3.3) and N tjtlog(t- »EN((t,oo);a,»)) =UN(t;a,n) (3.4) where C/jv (t; a,/*) satisfies the Jimbo-Miwa-Okamoto a form of the Pv differential equation (1.35) with uo =0, vi = -fi, v2= N + a, vz = N, (3.5) subject to the boundary condition *«^Nt. (3.6) Equivalently Uh'{t\a,n) is equal to the auxiliary Hamiltonian (2.12) with j = 1 and parameters (3.5). Proof. First observe that if {T[n]} satisfies the Toda equation (2.32) with f[0] = 1, then {<"cT[n]} is also a solution which is given by the determinant formula (2.43) with r[l](f) t-¥ tcf[l](t). Choosing c = —1/2 and substituting (3.1) for r[l](t) we thus have rn/2f[n] = det f^+*ifi(»i -t* + l,ws-«2 + 1;4)1 (3.7) where use has been made of the Kummer relation iFi(a,c;—t) — e~\Fi(c — a,c;t). Our first task is to use elementary row and column operations to eliminate the operator 5'+k in (3.7). Starting with elementary column operations, in column k {k — n — 1, n — 2,..., 1 in this order) make use of the identity ) (3.8) and add (vi — V2 + 1) times column k — 1. This gives n/2 i i+k 1 t~ f[n]ocdet\8 iFi(vi-v2 + l,V3-V2 + l;t) 5 iFi(vi - v2 + 2,v3 - v2 + 1; t) ... I o n_l. k = lt... ,n —1 17 Next, in column k (k = n — 1, n — 2,..., 2 in this order) make use of the identity (3.8) again and add (vi — v2 + 2) times column k — 1 to obtain t~n/2r[n] ocdet • ••"L=0>...jn_1 • (3.9) k=2,...',rt-lk=2'rtl Further use of (3.8) in an analogous fashion gives (3.10) j,k=0,...,n—1 At this stage we use elementary row operations to eliminate the operation (3-11) and subtract row j — 1 to get t~n/2f[n] oc det — V2 ,V3 — V2 Next, in row j (j = n — 1, n — 2,..., 2) in this order) make use of the identity (3.11) again and subtract 2 times row j — 1 to obtain rn/2f[n](xtdet — V2 + 1 + k, V3 — V2 + 3; t) 3=2 n-1 fcO 1 Further use of (3.11) in an analogous fashion gives n/2 n( 1)/2 r f[n] oc t "- det \(v3 -vi- k)jiFi{vi - v2 - v (3.12) L 2 j,k=0,...,n-l thus eliminating entirely the operation S. We now substitute the integral representation for 1F1 deducible from (2.41) to obtain n/2 1)/2 tu k+vl V2 t- f[n] oc - det [ P e u - (l - u)«3-»i-i+j-* dv\ IJO ij,k=O n-l n< n 1)/2 1 l 2 3 1 71 Ht 1 1 2 3 1 1 = t - - T dm c'» «J -" (l - MI)" -" - • • • T d«n e -< - ' (l - tin)" - ' -" Jo Jo Regarding the determinant in the last line of this expression, we have n-l n+:> k l k det [«*+i(l -uj+i) ~ ~ y _ = JJ(1 -uj+1y det[u +1]j,k=0,...,n-i, ' ' ,7—0 where the equality follows by adding the fc+l-th column to the k-th (k = n — 1,...,/) in a sequence of sweeps I — 1,... ,n— 1. All factors other than the determinant in the multidimensional integral are symmetric in {«,} so we can symmetrize this term without changing the value of the integral (apart from a constant factor n\ which is not relevant to the present discussion). Noting that n— L Sym( Y[{\ - :± 11(1* -«*)*, (3.13) 3=0 } 18 for some sign ±, and substituting in (3.9) gives (3.2). n/2 (n 1)/2 1 2 s vi n s r f[n] oc r ~ f dm e'^u; — (1 - Uly - - ••• f dun e^-tC^l - «»)' """" Jo Jo 3 r[N](t) ex r(o2 provided the parameters are given by (3.5). The result (3.4) now follows by substituting this result in (2.15) with j = 1. For the boundary condition (3.6), we see from (3.4) and (1.8) that ~ aN + N2 -t s->o where JN(O,, M)EJ=I A,-] denotes the integral (1.18) with an additional factor of SJLi A; in the integrand. Noting that X^j=i A? can be written as a ratio of alternants allows the integral to computed using the method of orthogonal polynomials and leads to the result (3.6). • Next we consider (2.43) with initial value (2.42). Proposition 6. Let f[n] be specified by the determinant formula (2.43) with l/2 f\\\{t) = t ip{yi - v2 + 1, v3 - V2 + 1; t). (3-14) Then we have 2 f°° c f°° i" i 1—r n tui V3 vl n Un vs Vl n f[n] oc t I duie~ Ui (l+ui) ~ ~ • • • I dune~ Un (l+un) ~ ~ TT (uk—Uj) . Jo ^° l It follows from this that "/ J\J \\yJ* Cji tl« Lt I — O t x iY II] 10-J.DJ and " =^(^0,^) (3.17) where Viv(£; a, /J-) is equal to the auxiliary Hamiltonian (2.12) with j = 1 and parameters (3.5), and so satisfies the Jimbo-Miwa-Okamoto form o/Pv (1.35) with parameters (3.5). The latter is to be solved subject to the boundary condition ,n) ~ -Nt + N{a - p) - ^Jl±£ll + O(l/t2). (3.18) t—>oo t Proof. Analogous to (3.7) we have n/2 j+k r f[n] = det \s i)(vi - v2 + 1, v3 - v2 + 1; t)] (3.19) L J j,k=O,...,n— 1 We now adopt the identical strategy as used in the proof of Proposition 5, with the identities (3.8) and (3.11) replaced by 5i/>(a, c;t) = a((a-c+ l)^(o + 1, c; t) - t(a, c; and —ip(a,c;t) = i(>(a,c;t)-ip(a,c+l;t) 19 respectively. The results (3.15) and (3.17) are then obtained by repeating the working which led to (3.2) and (3.4). For the boundary condition, we see from (3.17) and (1.7) that N aIN(u)[Y, , A,] where -TAKAOEJII A?) denotes the first integral in (1.18) with an extra factor of ]^jLi -\? m the integrand. This integral can be computed by changing variables Xj >->• eXj in the definition of /JV(A*), differentiating with respect to e, and setting e — 1. D 3.3 A relationship between transcendents The evaluations (1.47) for pmin(s;a) and (1.38) for Etf((0,s);a,0) substituted into (1.11) imply that ATATfl +l( a VN(t;a,2) = -(a+l + 2N)+t + VN+1(t;a,0)+t^ f' '^ (3.20) In our previous study [17] we encountered an analogous identity relating some Prv transcendents. Like (3.20), the Piv identity in [17] was discovered using a relation of the type (1.11). A subsequent independent derivation using Backlund transformations was also found and presented. Likewise the identity (3.20) can be derived from the formulas (2.26) and Table 1. An identical formula relates UN{t;a, 2) and UN+i(t;a,0) which can be understood from the evaluation of pmax(s; a) and EN((S, OO); a, 0). Proposition 7. Introduce the shift operators T2 and T3, which have the action on the parameters v specified by 3111 1131 T2-V = {VI + -,V2- -,V3--,V4--), T3-V=(V1- -,V2~ -,VS+-,V4- ~) (3.21) and consequently from (2.28) and Table 1 the representation T2 = S1TTS3S2, TZ = S2S1TTS3. (3.22) Then we have 1 r8r2- ViV+i(t;ol0) = VN(t;a,2) (3.23) and use of (2.26) and Table 1 on the LHS reduces this to (3.20). Proof. We take advantage of the arbitrariness in (2.13) which with {vi} given by (1.36) with N <-> N+ 1 allows us to write (i) VN+i{t;a,0) = cr (3.24) The actions (3.21) then imply ( T3T2-VAr+1(t;a,O) = (r 2i+ v3— v\=—2 which recalling (1.39) and (2.13) is the equation (3.23). X To evaluate the action of T3T2~ on the LHS of (3.23), first note from (2.12) and the final equation in (2.19) that for general parameters (1) <7 = tH + (V3 - Vl)(«2 - t>l). We remark that the simplifying feature of arranging the parameters so that «2 = «3 — i>i = 0 in (3.24) is that (2.18) takes the reduced form tH = PUii - l)(p + t) - (<*i + a3)q + ail. (3.25) L Ct*22=0 J 20 1 The explicit formula (3.25) will become important later on. For now see seek the action of T3T2 on a^ for general values of the parameters. First we note that the generators of the extended type A3 afiine Weyl group possess the algebraic properties (2.20) which can be used in the formula for TzT^1 implied by (3.22) to show that T3T2 = S2S3S1S0S1S3. Now, it follows from (2.26) and Table 1 that S0S1S3 (tH) = tH + ao-^— +00(02-1) + (ai+ao)t. (3.26) p + t Let us now consider separately the second term in (3.26). Use of Table 1 shows 3 U °p + t> q(q Furthermore, we see from Table 1 that in the special circumstance tq(q - p + t q(q - l)(p + t) - ai(q — 1) — tH where the final equality follows from (3.25) and (2.18). For the remaining terms in (3.26), we see from (2.26) and Table 1 that S2S3S1 [tH + 00(02 — 1) + (oi+oo)i] — tH + t — a.o- 012=0 The above results together with the simple formula imply = tH (3-28) 1)3—^1=0 tii ^3—vi =0 Substituting the values of V2 — v\ and VA — i>i from (3.24) gives (3.20). • 3.4 Difference Equations We know the logarithmic derivatives of the r-functions C/jv(t;o,p) and V}v(£;a,p) satisfy the second order second degree differential equation (1.35). Here we will utilise the Schlesinger transformation theory to show that they also satisfy third order difference equations in both a and fi variables. Proposition 8. The logarithmic derivative UN(t;a,fi) satisfies a third order difference equation in the variable a ~ _ ] [U-U~N\ -Hu -u;--= J- N(a+fi+l+t) + U- (a+l)(U - U) {U - U+a+fi) \Nfj, + (N+a+fj,+l)U - (N+a+fi)u] +1 \fiN - yXJ + (N+a+fi)U - (N+a)ll] L J ^ i- (3.29) where U := UN{t;a,fi),U := f/jv(*;o — l,fi),U := UN(t;a + l,fi), etc and the boundary conditions are expressed by C/jv(i;a,/u) at three consecutive a-values for all N,t,/i. In addition UN(t',a,iJ,) satisfies a third order difference 21 equation in \i -t(U-U) = {U - U+2N+a+fi+l) \-N(N+a) + (N+ft+l)U - (N+fi)u\ -t \-N(N+a) + (ft+l)U - (N+n+l)U + NU -N(2N+a+fj,+ l-t) + U- {N+n+l){U - U) (U - U+2N+a+pi) \-N(N+a) + (N+a+fi+l)U - (N+a+rfu) - t \-N(N+a) - ftU + (N+a+ft)U - (N+a)u\ -(N+a)(2N+a+n-t) + U- (N+a+fi){U - U) (3.30) *— —> where now U := UN(t',a,fi — 1),U := UN(t;a,fi+ 1). Proof. The difference equation generated by TJ"1 (2.47) with the parameter identification V2 — v\ — —p, v$ — v\ = N + a, V4 — v\ = N is not directly useful as this leads to a difference equation in both iV and a (there are no difference equations in the parameter N alone - only combined ones with a or fi). However difference equations generated by the other shift operators can be simply found using this one with a permutation of the parameter identification. Thus for the difference equation in a one can use V2 — v\ = —/J,, v& — v\ = N, Vi — v\ = N + a and the relation tH = UN{t;a,fi) + fJ-N which gives the result is (3.29). The second result (3.30) follows from the parameter identification «2 — «i = N, V3 — v\ = N + a, VA — vi = —fi and tH = UN (t; a, ft) — N(N+a). • Both of these difference equations are of the third order and linear in the highest order difference and it may be possible to integrate these once and reduce them to second order equations, however we do not pursue this question here. Although we have stated the difference equations for UN{t; a, fi), it is clear that V)v(t; a, ft) satisfies these as well although subject to different boundary conditions. 3.5 EN((0,s);a,fj) for aeZ+ In a previous study [16] the quantities EN((Q,s);a,0) and EN((0, S); a,2) for a € Z were expressed in terms of a x a determinants. This was done using the method of orthogonal polynomials to simplify the corresponding multiple integrals. In fact we can easily express EN((0, S); a, fi) for a 6 Z+, with fj, general, as an a x a determinant using the methods of the present study. Proposition 9. For a € Z the function at \ I Jj,fc=0,...,a-l 4([^^]^=o ^) (3.31) Consequently, for a 6 Z>o, Ns EN((0,s);a,fi) Proof. Choose vz — vi — —N, Vi — V2 = n (3.33) in (3.1). Then according to (2.43), for a 6 Z>i t-a/2f[a] = det [>+VxFi (-N, ft + 1; -tj\ . (3.34) L lj,k=0,...,a-l Using the fact that 22 where L^ denotes the Laguerre polynomial, substituting (3.34) in (2.44) and then substituting the resulting expression in (2.15) gives the first formula for cr^ in (3.31). In specifying the parameters in (2.15) we have made use of (3.33), the fact that v^ — v\ = a, as well as (2.2). The theory noted in the sentence containing (2.13) tells us that a(3) satisfies (1.35) with vo —0, v\ = V2 — V3 — —\x, v To obtain the second equality in (3.31) we make use of the identity 5 [L% (-t)e'] = [(N + 1 in conjunction with elementary column operations, proceeding in an analogous fashion to the derivation of (3.9) using the identity (3.8). This shows det r^+VL^-t))] oc det [*VL&+Jk (-*))] L J j,k=O,...,a — 1 L J j,k~O,...,a~ 1 The second equality in (3.31) now follows by applying to this the general identities j j det \5 (u(t)Mt)j\ = («(<))" det \s fk(t)] lj,k=O,...,a-l idP -ij,k=O a-1 The function in the logarithm of the second equality in (3.31) is of the form t~N>xe~Nt times a polynomial in t. But for a € Z>o, we know from the second integral formula in (1.7) that t~N^E((0,t);a,fi) has the same structure. Since a^ also satisfies the same differential equation as Vw(£; a,n) in (3.17), the formula (3.32) follows. • The determinant formula (3.34) can be written as an a-dimensional integral by using (3.2). Because (3.33) implies an exponent — (N + a) for the factors (1 — Uj) in the integrand, we must first modify the interval of integration. For this purpose it is convenient to first change variables UJ H-> 1 — it,. Then instead of the interval of integration [0,1] we choose a simple, closed contour which starts at Uj — 1 and encircles the origin. In particular, choosing this contour as the unit circle in the complex Uj plane gives ra/2f[a] oc : \ -H- ~ ' " ' / CUECT Consequently we have the following generalization of (1.37). Proposition 10. For a £ Z>o, 1 2 N EN((0, s);a>ti) = *-"'££;$) { fr + e~ ^) (l + e^Te"™"' )^ (3.35) We remark that the identity (3.35) is itself a special case of a known more general integral identity [11]. The latter identity involves the PDFs 2^fc_e2«^.p (_i/2 < a;, < 1/2) defining what we will term the ensembles L^E and C/3E respectively. For /? = 2, these PDFs were introduced in (1.4) and (1.30) as the LUE and CUE. For {3 = 1 and 4 these PDFs also have a random matrix interpretation. 23 They correspond to the case of an orthogonal and symplectic symmetry respectively, and give rise to the matrix ensembles LOE, COE and LSE, CSE. Now define 1=1 (note that this reduces to (1.7) for /? = 2). Then it follows from results in [11], derived using the theory of certain multi-variable hypergeometric functions based on Jack polynomials, that for a G Z>o 'C(4//3)Ea where M^\a,b) denotes the integral (1.28) with exponent 2 in the product of differences replaced by /3. 4 r-function theory of Pm and hard edge scaling In the Laguerre ensemble the eigenvalues are restricted to be positive. For N large, the spacing between the eigenvalues in the neighbourhood of the origin is of order 1/N. By scaling the coordinates as in (1.50) the eigenvalue spacing is then of order 1 and well defined distributions result. In this limit the Pv system degenerates to the Pin system so it is appropriate to revise the r-function theory of the latter. 4.1 Okamoto r-function theory of Pni Following [30], the Hamiltonian theory of Pm (actually the Pup system rather than the Pni ) can be formulated in terms of the Hamiltonian 2 2 2 tH = g p - (g + wig - t)p + -(vi + v2)q. (4.1) Thus by substituting (4.1) in the Hamilton equations (2.3) and eliminating p we find that y(s) satisfies the Pni differential equation ~TT — — ("T~) r~ H—(ay2 + P) + 72/ H— ds^ y \ds / s ds s y with q(t) = sy(s), t = s2 and a = — 4v2, P = 4(«i + 1), 7 = 4, 5 = — 4. Note that with H specified by (4.1), the first of the Hamilton equations (2.3) gives tq = Iq p — {q + v\q — t). (,4.zj Thus, by using this equation to eliminate p in (4.1), we see that tH can be expressed as an explicit rational function of q and q'. Analogous to Proposition 1, it is straightforward to show that tH plus a certain linear function in t satisfies a second order second degree equation. Proposition 11. With H specified by (4-1), define the auxiliary Hamiltonian h = tH+^y\-\t. (4.3) The auxiliary Hamiltonian h satisfies the differential equation (th")2 + viV2h' - (4(h')2 - l)(ft - tti) - \{v\ + v%) = 0. (4.4) 24 Vl V2 P q t -1 — V2 -1-V! 1 -\{vi-' t t so t [«(P~1) J2)] + 1 q Si V2 Vl P q + t 2(P -7) S2 Vl -V2 1-p -q -t Table 2: Backlund transformations relevant to the Pni Hamiltonian (4.1). Proof. Following [30], we note from (4.1) and the Hamiltonian equations (2.3) that ,, 1 h=p-- (4.5) th" — 2(1 — p)pq + vip — -(vi + v2). Using the first equation to substitutth"e fo r-Viti p in the second gives th" - viti + \v2 H §(l-4(/i')2) ' w (1-2A') while we can check from (4.3), (4.1) and the first equation in (4.5) that h-tti = (qp - ~vif - q[qp - -(vi + v2)]. Substituting for qp and q, and simplifying, gives (4.4). • Of interest in the random matrix application is the variant of (4.4) satisfied by A straightforward calculation using the result of Proposition 11 shows that am satisfies (tam) ~ viv2(ain) + C///(4cr/// — l)(crj// — tcrjjj) — ~r^(vi — V2) = 0. (4.7) Note that with the r-function defined in terms of the Hamiltonian (4.1) by (2.14), we have (4.8) 4.2 Backlund transformations and Toda lattice equation For the Hamiltonian (4.1), Okamoto [30] has identified two Backlund transformations with the property (2.16): Tl -V= (Vl +1,1)2 + 1), T2-V=(«l+1,«2-1). (4.9) The operators Ti and T2 can be constructed out of more fundamental operators so,Si,S2 associated with the underlying B2 root lattice, whose action (following [30] and [25]) on v,p, q is given in Table 2. According to Table 2 we have = SQS2SIS2, T2 = S2S0S2S1. (4.10) Analogous to the situation with the Hamiltonian (2.30) in the Pv theory, introducing the sequence of Hamil- tonians by H[n] := H (4.11) where H is given by (4.1), the operator Ti can be used to establish a Toda lattice equation for the corresponding r-function sequence (2.31). 25 Proposition 12. [30, 25] The r-function sequence corresponding to the Hamiltonian sequence (4-H) obeys the Toda lattice equation where f[n]:=r2/2r[n](i/4). (4.13) Proof. Analogous to (2.34) and (2.35) we have S log ^"-IM^ + I] = q[n}(l-p[n}) - Tf^nKl - Tf^n]). (4.14) 1 Making use of (4.10) we can use Table 2 to explicitly compute T^gln] and T1~ p[n]. This shows 2 g[n](l - p[n}) - T^q[n]{l - Tf^M) = -^(2q[n]P [n] - (2q[n] + vi)p[n] + |(«i + t But according to (4.1) and the second equation in (4.5), this latter expression is equal to Jlog(J^iiir). Substituting in (4.14) we deduce that - l]r[n Substituting for r[n] according to (4.13) and taking C = 1 gives (4.12). • 4.3 Classical solutions Okamoto [30] has shown that for the special choice of parameters vi = —V2, the Pni system admits a solution with T[0] = 1, and allows r[l] to be evaluated as a Bessel function. Proposition 13. [30] For the special choice of parameters Vi = -V2 (4.15) in (4-1) it is possible to choose T[0] = 1. The first member r[l] of the r-function sequence corresponding to (4-11), after the substitution 11-» i/4, satisfies the equation t(f[1])" + («i + l)(f[l])' - if[l] = 0, f[1] := r[l](t/4). (4.16) For vi £ Z, iAis equation has the two linearly independent solutions in terms of Bessel functions vl/2 T[l]=t~ I±vl(Vi) . (4.17) (/or «i 6 Z, 7W1 and 7_V1 are proportional and (4-11) only provides one independent solution). Proof. Substituting (4.15) into the definition (4.1) of H we see that it is possible to choose p = 0, H = 0, (4.18) the latter allowing us to take T[0] = 1. To calculate g we use the Hamilton equation . / dtH = -{q2 + viq-t). (4.19) tq = -s— j>=0 Now, it follows from (2.28) with H given by (4.1) and T by T\ that Substituting this in (4.19) and changing variables 11-> i/4 gives (4.16). • 26 For definiteness take the + sign in (4.17) and let v\ = v. This substituted in (4.13) with n = 1, and the corresponding formula for f [1] substituted in (2.43) shows that ( 1)/2 t" "- f[n] = det[ Proposition 14. We have + ( 1)/2 det[^ */,,(V*)k*=o »-i = (t/4)" "- det[J,_*+I,(V*)fc,*=o «-i. (4.21) Proof. Let t = s2 and note that did 1 to conclude j+k ( 1) +k det[S Iv(V~t)]j,k=o B-i = 2-" "- det[8i ll/(s)]j,k=o,...,n-i- (4.22) We now adopt a very similar strategy to that used in the proofs of Propositions 5 and 6. Briefly, we first make use of the identity SsIu(s) = slv+i(s) + ulv(s) together with elementary row operations to eliminate the operator 81 in (4.22), obtaining + k j det$ *J,,(8)fc,*=o »-i = det[8 s lu+j{s)]jtk=o n_x. (4.23) Next, use is made of the identity j j+1 j 5ss h+j(s) = s h+j-i(s) - vs lv+j{s) (4.24) to perform elementary column operations thus reducing the RHS of (4.23) to the form j +1 h+j(s) tf-V /,,+J--i(«)],-=o „-!. (4.25) fc l 1 Next we apply elementary column operations, using the identity (4.24) with j <-¥ ji + 1, v H-> V — 2, to show that (4.25) is equal to Continuing in this fashion, using the identity (4.24) with appropriate substitutions, reduces the RHS of (4.23) to Substituting in (4.22) gives (4.21). • Corollary 1. The function 2 2 («i,i>2) = (v + n, -v + n) (4.27) and boundary condition 27 Proof. It follows from Proposition 14, (4.20) and (4.13) that n /2 r[n](t/4) oc r " det[I,--fc+v The equation satisfied by aui(t) follows by substituting this, and the parameter values (4.27), in (4.8). To obtain the boundary condition, we make use of the well known Toeplitz determinant formula det iU k)ede /( ?,|2 [hfj^- ~ \^0^=^/>^>-V> ^ n together with the integral representation ine+z cose In(z) = — T e~ M, neZ 27r J—nr to rewrite the determinant in (4.26) as a multidimensional integral, det[Ij-k+v(Vi)]j,k=o,...,n-i 00 91 yieM<> iv9 =~j7^- rde^^ * -^••• rd0ne '- ' n ^-e^f (4.29) n n.(2n) J_K J_lr K-^K-,, valid for v £ Z. For large t the dominant contribution to the above integral comes from the neighbourhood of dj — 0 (j = 1,... ,n). Expanding to leading order about these points and changing variables shows n t n det[Ij-k+i.(Vi)]jlk=o,...,n-i ~ Ce t~ ' . t—>oo Substituting this in (4.26) gives (4.28). • The result of Corollary 1 is relevant to the hard edge scaling of EN((0, S); a,/i), which gives the quantity J5hard(i; a,fi) defined by (1.65). Now, in an earlier study [16], it was shown that for a G Z>o hard /4 S (i;a,2).oc e-* *-° det[7,-_*+2(V«)],-,fc=o>...,«-i (4.30) Thus, as already noted in [16] in the case of £?hard(t; a, 0) (deduced from knowledge of (1.66)), it follows from Corollary 1 that Shard(i;o,/i) for ft = 0 and fi = 2 can be characterized as the solution of the equation (4.7) with parameters (vi, V2) = (a + fj,,a — /J,). The results (4.30) were obtained by computing the limit of the RHS of (3.32) using the asymptotic formula x/2 a/2 a a/2 1/2 e~ x L N(-x) ~ N Ia{2(Nx) ). The same approach allows £'hard(t;o,/^) to be computed for general p, and a € Z>o, giving the result hard /4 Ml/2 JB (t;a,Ai) oc e-' r' det[J,-_*+/i(^)fc,*=o,...,.-i. (4.31) It then follows from Corollary 1 that the result (1.69) holds for general ft and a € Z>o- For general o > — 1 the result (1.69) can be deduced from the first formula in (1.38). This task will be undertaken in the next subsection. To conclude this subsection we make two remarks. The first is that substituting (4.29) in (4.31) shows that for \i 6 Z and a G Z>o (432) 28 This identity, relating an average in the infinite LUE scaled at the hard edge to an average in the CUE of dimension a, has previously been derived [11] as a scaled limit of (3.35). With c := 2(fj, + 1)//? and fj, such that c 6 Z>o, it was also shown that the identity (3.36) in the hard edge limit reduces to yIras 4 d9ae '"-*-* f] |c«»-e"i| "» (4.33) where A r(i 11 and E^haid, like Eh&rd specified by (1.65), is normalized so that it equals unity for t = 0. As a second remark, we note that a feature of the differential equation (4.4) is that it is unchanged by the mapping t H> — t, V2 i-> — «2- Using this, it follows from (4.3) and (4.6) that (4.7) is also satisfied by where the equality follows from (4.8). But by an appropriate modification of Propositions 13 and 14, a determinant formula for r(—£/4) can be given for certain («i, «2). Performing these modifications and substituting the resulting formula in (4.34) leads us to the conclusion that 4/ 2/2 -tj log (e~ V tet[Jj-k+»b/i)]i,k=o,...,n-i) (4.35) satisfies (4.7) with parameters •n,u — n). In the n = fj, = 2 and v = a case, substituting (4.35) for 4.4 Schlesinger Transformations Integrable difference equations are found to arise from the Schlesinger transformations generated by Ti,Ti [33, 32, 26] acting on q and p and we consider these here, along with the difference equations for the Hamiltonians. Proposition 15. [32] The Schlesinger transformations of the Pm system for the operators Ti(T2) generating the parameter sequences (vi + n, V2 4- n) (respectively (vi + n, V2 — n)) acting on the transcendent q correspond to second order difference equations of the alternate discrete Painleve II, d-Pn , equation (respectively its dual) 1 vi + V2 + 2 + 2re 1 vi +V2 + 2n _ _i q V2 + n ,..., 2 q[n]q[n + l]+t 2q[n-l]q[n]+t~q t t ' ( } \v2-v\-2-2n 1 V2 -vi -2n _ _t q vi +n (A-\7\ 2 q[n]q[n +1] -1 2 q[n - l]q[n] -1 ~q t t ' K ' Further, under the action of Ti, tH[n + 1] — tH[n] := q[ri\(l — p[n]) and this satisfies the second order difference equation -p)[n] + q{l-p)[n - 1] + |(«i - v,)] p)[n] +q(l-p)[n- while under the action ofT2, tH[n + 1] — tH[n] := 3[n]p[n] and this satisfies the second order difference equation X r 1 I" r 1 ( J- ^ [qp[n + 1] + qp[n] - |(«i + v2)] [qp[n] + qp[n -l]-±(Vl+ y2)] Win] [Mn] --(v1+ ,2)j = -t [qp[n+1]+qp[n]_vi_1_n][qp[n]+qp[n_1]_vi_n] • (4.39) 29 Proof. In the case of the T\ transformation the action on the canonical variables in the forward and reverse directions are t k{vi + v2 + 2 + 2n)t """ + ^DH—•»»+' p[n + 1] = |[«(p - 1) - \(vi - v2)} + 1 (4.41) i[n - 1] = — t—— (4.42) 5(1*1 +v2 + 2rc) P q ^\±\ (4.43) By combining (4.40) and (4.41) into the form r • i t Uvi+v2+2 + 2n) q[n + l] + -n = -^ j-—— 4.44) q[n] p[n + l] and eliminating p between this equation and (4.42) we have the result (4.36). The difference equation for the Hamiltonian can be found by solving A q(l-p)[n + 1] + q(l-p)[n] + \{v, - v2) = -\{v, + v2 + 2 + 2n) ./Vl^ ^ TIZ ^ ^ for q and X M q{l-p)[n - 1] +q(l-p)[n] + i(«i - t*) = i(«i + «2 + 2n) ^ (4.46) for p and then reforming g(l — p). The result is (4.38). The corresponding results for the T2 sequence can be found in a similar manner. • 4.5 Hard edge scaling i?hard(£; a, n) According to (1.65) and (1.38) we have for general a > —1 hard JB (s; a, n) = exp [ v(t; a, /j) dt (4.47) Jo where v(t;a,fj,)= lim fVN (t/4N; a, fi) + /J,N). (4.48) In this limit the differential equation (1.35) characterizing the Pv auxiliary Hamiltonian (2.12) and specifying VN degenerates to the differential equation (4.4) characterizing the Pin auxiliary Hamiltonian (4.3) thus identifying v with this quantity. Proposition 16. The function v specified by (4-48) satisfies the differential equation 2 {tv"f -(fj, + a) V)2 - v'{4v + l)(v - tv) - |(/i + a)v - ^ = 0. (4.49) Consequently v(t; a,/j) = - (am (t) + fi(fi + a)/2) (4.50) where am is specified by the Jimbo-Miwa-Okamoto a-form of the Painleve equation (1-67) with parameters (1-70) vi = a + fi, v2 = a — y, and is subject to the boundary condition (1-71) atl/2 v{t;a,n)t~ -\t+\ - \a{a + 2n). (4.51) 30 Proof. Making the replacement a i-> a — Nfj, in (1.35) with parameters (3.5), changing variables t >-» t/4N, o~(t/4N) i-> v(t), and equating terms of order N2 (which is the leading order) on both sides gives (4.49). Sub- stituting (4.50) in (4.49) shows am satisfies (167) with parameters as specified. The boundary condition is the same as (4.28) with n = a, v = ft. • Proposition 17. The function v(t;a,fi) satisfies a third order difference equation in the variable a (suppressing the additional dependencies) -a(a+ l)t = [v(a+l) - v(a-l)+a+fi][v(a+2) - v(a)+a+/j.+l] \-t - av(a+l) + (a+l)v(a)\ , (4.52) and a third order difference equation in \i -i, = Proof. These two results follow from applying the hard edge scaling form (4.48) to the finite-iV difference equations (3.29) and (3.30). Note that (4.53) is precisely the result that would be inferred from the Pm difference equation generated by the I2 shift (4.39) with the correspondence v(t;a,fj,) = tH -t/4. (4.54) ti->t/4 D We have noted in (1-73) that a corollary of Proposition 16 is the evaluation of pmin(s; a) in terms of v(t; a, 2). Since, analogous to (1.11), h d piTn(s;a) = -££ " ((0,s);a,0), (4.55) the results (1.73) and (1.66) substituted into this formula imply an identity between transcendents analogous to (3.20). As is the case with (3.20), this identity too can be independently verified. Proposition 18. With a(t) denoting the auxiliary Hamiltonian (4-6) with parameters (v\,V2) = (a + 2,a — 2), and aB(t) denoting the same quantity with parameters («i,«2) = (a,a), the identity (4.56) UB\l.) holds. Proof. The indirect derivation of this result has been sketched above. A direct derivation can be given by using the properties of the shift operator T2 in (4.9) and (4.10). First, denote by ii7[0] the Hamiltonian tH in (4.6) (and similarly define p[0], q[0]) so that aB(t) = ~(tH[0]) +t/4. (4.57) It then follows from the definitions of T2 and a(s) that a(t) = -(tT%H[0}) -{a+ 2) +1/4. (4.58) tt-it/i On the other hand, from the definition (4.1) and the property (2.16) of T2 we have that tT2H[Q] = tH[Q] - #]p[0] - (T2q[0])(T2p[0]). But from (4.10) and Table 2 it follows that T2q[0] = [q\fS\ ~ q[0](q[0]p[0]-a) + t) and thus tTlH[0] = tH[0] - o - 1 +1^^- (4.59) O-B(t) where to obtain the final term use has been made of (4.1), (4.6) and the analogue of the first equation in (4.5). Substituting in (4.58) and recalling (4.57) gives (4.56). D 31 4.6 The spacing probability p2(0; s) The evaluation (4.47) has consequence regarding the nearest neighbour spacing distribution P2(O; s) for the scaled, infinite GUE in the bulk. This quantity has a special place in the Painleve transcendent evaluation of gap probabilities because it motivated the study of the probability E2(Q; s) of no eigenvalues in an interval of length s in the scaled, infinite GUE through the relation and £2(0; s) in turn was the first quantity in random matrix theory to be characterized as the solution of a non-linear equation in the Painleve theory [22]. More recently, building on the evaluation of i?2(0; s) from [22], it has been shown that [18] ) ^r^dt (4-60) s where cr(s) is specified by the solution of the non-linear equation s2(cr")2 + 4(sff - Z)(sa -a + (a)2) - 4(a')2 = 0 (a close relative of (1.35) for a particular choice of the parameters) subject to the boundary condition 2 2 2 ( f[(A - ; A, ^/2iV it follows by replacing s by rcs/y/2N, making use of the definition (1.65), and the property P2(O; s) ~ \s2 which follows from the boundary condition for CT(S)i n (4.60) (this fixes the proportionality constant) that d 2 h! d 2 P2(0;2s) = ^tE^ {{-Ks) ; \,2)E " {{ixs) ;-\,2). (4.62) According to (4.47) this specifies P2(O; s) in terms of Painleve III transcendents whereas (4.60) involves Painleve V transcendents. 5 Concluding remarks The results of the paper have already been summarized in Section 1. Here we want to draw attention to a feature of this work (and our previous work [17]) which requires further study. This feature relates to the boundary conditions in the scaled limits, in particular the specification of £?hard(s; a, ft) by (1.69). One observes that the boundary condition (1.71) is even in /j,, as is the differential equation (1.67) with parameters (1.70). Thus according to this specification the solution itself must be even in /J,, but this contradicts the small t behaviour which for the formula (1.69) to be well defined must be t0\ a) + O(f), (e>0). (5.1) The situation is well illustrated by the case a = 1, for which (4.26) shows a(t) = -t| log (e-*> V2'%(^)). (5.2) 32 The small t expansion of I^y/t) shows 0) in agreement with (5.1), whereas the large t expansion of /M(\/i) shows where the at are even functions of fi. Thus the asymmetry between /j, and — n can only be present in an exponen- tially small term for t —> oo. This is clear from the exact expression \ (5.5) Furthermore, although the boundary condition (5.1) distinguishes the cases ±fj,, the differential equation (1.67) also requires that the leading (in general) non-analytic term also be specified for the solution to be uniquely specified (for example, with /j, = 0 this term is proportional to ta+1). A practical consideration of this discussion is that the boundary condition (1.71) does not uniquely determine the solution of (1.67), so i?hard(.s;a,^) is not uniquely characterized. The same remark applies to the formula (1.56) for Eso{t(s;n). Indeed the inadequacy of the boundary condition (1.58) to uniquely determine the solution is already apparent in the our discussion of the formula (1.59). Acknowledgements This work was supported by the Australian Research Council. References [1] M. Adler, T. Shiota, and P. van Moerbeke. Random matrices, vertex operators and the Virasoro algebra. Phys. Lett. A, 208:67-78, 1995. [2] M. Adler and P. van Moerbeke. Integrals over classical groups, random permutations, Toda and Toeplitz lattices. math.CO/9912143, 1999. [3] V.E. Adler. Nonlinear chains and Painleve equations. Physica D, 73:335-351, 1994. [4] J. Baik and E.M. Rains. The asymptotics of monotone subsequences of involutions, math.CO/9905084, 1999. [5] T.H. Baker and P.J. Forrester. Random walks and random fixed point free involutions. Preprint, 2000. [6] T.H. Baker, P.J. Forrester, and P.A. Pearce. Random matrix ensembles with an effective extensive external charge. J. Phys. A, 31:6087-6101, 1998. [7] A. Borodin and P. Deift. Fredholm determinants of a class of integrable operators are Jimbo-Miwa-Ueno tau-functions. In preparation, 2000. [8] CM. Cosgrove. Chazy classes IX-XI of third-order differential equations. Stud, in Appl. Math., 104:171-228, 2000. [9] CM. Cosgrove and G. Scoufis. Painleve classification of a class of differential equations of the second order and second degree. Stud, in Appl. Math., 88:25-87, 1993. [10] P.J. Forrester. Log-gases and Random Matrices. Book in preparation. [11] P.J. Forrester. Exact results and universal asymptotics in the Laguerre random matrix ensemble. J. Math. Phys., 35:2539-2551, 1993. [12] P.J. Forrester. The spectrum edge of random matrix ensembles. Nucl. Phys. B, 402:709-728, 1993. 33 [13] P.J. Forrester. Normalization of the wave function for the Calogero-Sutherland model with internal degrees of freedom. Int. J. Mod. Phys. B, 9:1243-1261, 1995. [14] P.J. Forrester. Inter-relationships between gap probabilities in random matrix theory. Preprint, 1999. [15] P.J. Forrester. Painleve transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles, nlin.SI/0005064, 2000. [16] P.J. Forrester and T.D. Hughes. Complex Wishart matrices and conductance in mesoscopic systems: exact results. J. Math. Phys., 35:6736-6747, 1994. [17] P.J. Forrester and N.S. Witte. Application of the r-function theory of Painleve equations to random matrices: PIV, PII and the GUE. Preprint, 2000. [18] P.J. Forrester and N.S. Witte. Exact evaluation of the spacing distribution for random matrix ensembles in the bulk. Lett. Math. Phys., 53:195-200, 2000. [19] P.J. Forrester and N.S. Witte. r-function evaluations of gap probabilities in orthogonal and symplectic matrix ensembles. Preprint, 2001. [20] L. Haine and J.-P. Semengue. The Jacobi polynomial ensemble and the Painleve VI equation. J. Math. Phys., 40:2117-2134, 1999. [21] M. Jimbo and T. Miwa. Monodromony preserving deformations of linear ordinary differential equations with rational coefficients II. Physica, 2D:407-448, 1981. [22] M. Jimbo, T. Miwa, Y. Mori, and M. Sato. Density matrix of an impenetrable Bose gas and the fifth Painleve transcendent. Physica, 1D:8O-158, 1980. [23] K. Johansson. Shape fluctuations and random matrices. Commun. Math. Phys., 209:437-476, 2000. [24] I.M. Johnstone. On the distribution of the largest principal component. Preprint, 2000. [25] K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, and Y. Yamada. Determinant formulas for the Toda and discrete Toda equations, solv-int/9908007, 1999. [26] F. Nijhoff, J. Satsuma, K. Kajiwara, B. Grammaticos, and A. Ramani. A study of the alternate discrete Painleve II equation. Inverse Prob., 12:697-716, 1996. [27] M. Noumi and Y. Yamada. Higher order Painleve equations of type A^. Funkcial. Ekvac, 41:483-503, 1998. [28] K. Okamoto. Studies of the Painleve equations. III. Second and fourth Painleve equations, Pn and Piv • Math. Ann., 275:221-255, 1986. [29] K. Okamoto. Studies of the Painleve equations. II. Fifth Painleve equation Py • Japan J. Math., 13:47-76, 1987. [30] K. Okamoto. Studies of the Painleve equations. IV. Third Painleve equation Pm. Funkcialaj Ekvacioj, 30:305-332, 1987. [31] E.M. Rains. Increasing subsequences and the classical groups. Elect. J. of Combinatorics, 5:#R12, 1998. [32] A. Ramani, B. Grammaticos, T. Tamizhmani, and K.M. Tamizhmani. On a transcendental equation related to Painleve III, and its discrete forms. J. Phys. A:Math. Gen., 33:579-590, 2000. [33] A. Ramani, Y. Ohta, and B. Grammaticos. Discrete integrable systems from continuous Painleve equations through limiting procedures. Nonlinearity, 13:1073-1085, 2000. [34] C.A. Tracy and H. Widom. Fredholm determinants, differential equations and matrix models. Commun. Math. Phys., 163:33-72, 1994. [35] C.A. Tracy and H. Widom. Level-spacing distributions and the Bessel kernel. Commun. Math. Phys., 161:289-309, 1994. [36] C.A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Commun. Math. Phys., 177:727-754, 1996. 34 «* [37] C.A Tracy and H. Widom. On the distributions of the longest monotone subsequences in random words. math.CO/9904042, 1999. [38] C.A. Tracy and H. Widom. Random unitary matrices, permutations and Painleve. Commun. Math. Phys., 207:665-685, 1999. [39] H. Watanabe. Defining variety and birational canonical transformations of the fifth Painleve equation. Analysis, 18:351-357, 1998. [40] E.T. Whittaker and G.N. Watson. A Course of Modern Analysis. CUP, Cambridge, 2nd edition, 1965. [41] N.S. Witte, P.J. Forrester, and CM. Cosgrove. Gap probabilities for edge intervals in finite Gaussian and Jacobi unitary matrix ensembles. Nonlinearity, 13:1439-1464, 2000. 35