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... is E hard 1 (s; a), the probability that the interval (0, s) is free of eigenvalues. This was evaluated in terms of a Painlevé V transcendent in [19] and expressed as a τ-function for a PV system in =-=[30]-=-. The latter result can be written where satisfies (4.8) with t ↦→ 2x and d log Ehard 1 (x dx 2 ;(a − 1)/2) = ˜ hV (x) (5.37) σV (x) := x ˜ hV (x) + 1 4 x2 − On the other hand, if follows from Proposi...

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...nal distribution is defined by summing over every second row of the semi-standard tableaux. We also draw attention to the fact that FGUE(s) and FGOE(s) are τ-functions for certain Painlevé II systems =-=[37, 19]-=-. Similarly, FGSE(s) is the arithmetic mean of two τ-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition [19]. Refer...

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...e PII equation (2.9) )) q(t)dt . (2.10) (H ′′ II )2 + 4(H ′ II )3 + 2H ′ II (tH′ II − HII) − 1 1 (α + 4 2 )2 = 0. (2.11) Introduce the auxiliary Hamiltonian and the corresponding τ-function Then from =-=[10]-=- we know that hII(t;α) := HII(t;α) + t2 8 τII(s;α) = exp ( − ∫ ∞ s (2.12) ) hII(t;α)dt . (2.13) E soft 1 (0;(s, ∞)) = τ + II (s;0) (2.14) E soft 2 (0;(s, ∞)) = τ + − II (s;0)τII (s,0) (2.15) E soft 4 ...

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...N +1)/2⌋)E2(0; J + ; y 1/2 g2(y 1/2 ); ⌊N/2⌋) , (J+ is the positive member of a pair of intervals composing J). Examples of where this relation has been useful can be found in the above reference and =-=[8]-=-. An example of such a double interval statistic is one arising from an ensemble of random unitary N × N matrices with the joint eigenvalue pdf (3) p(z1, . . . , zN) = CN,a N∏ l=1 ∏ 2a |1 − zl| 1 1≤j<...

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...racing its behavior close to zero, but letting n going to infinity first is well studied (see [4, Ch. 3] and the references therein). For finite ensembles, it has been shown by Forrester and Witte in =-=[11]-=- 6 ANTONIO LERARIO AND ERIK LUNDBERG that fβ,n can be evaluated using methods from integrable systems. For example, if β = 1 and n is even:1 f1,n(ε) = τσV (ε 2), where τσV is a function satisfying: σV...

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...e function: pµ(ε) = P{σi1(Q)2 + · · ·+ σiµ(Q)2 ≤ ε2 for some 1 ≤ i1, . . . , iµ ≤ n}. In the case µ = 1 the function p1(ε) is called the gap probability and has been widely studied (see, for instance =-=[12, 14, 19]-=-). Using the exclusion-inclusion principle we see that we can write: (18) pµ(ε) = n∑ k=µ (−1)k−µ ( n k ) P{σ1(Q)2 + · · ·+ σk(Q)2 ≤ ε2}︸ ︷︷ ︸ gk(ε) (the binomial coefficient comes from the fact that t...

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...E(n) is called gap probability ; we consider this probability as a function of ǫ and denote it by fn(ǫ). In the case n is even, fn(ǫ) can be evaluated using methods from integrable systems. Following =-=[7]-=- we have:3 (6) fn(ǫ) = τσV (ǫ 2), where τσV is a function satisfying: (7) σV (t) = t d dt log τσV (t) and lim t→0+ σV (t)t −1/2 = − Γ( n+1 2 ) Γ(n2 )Γ( 1 2 )Γ( 3 2 ) = −cn. We denote by Γ the Euler Ga...

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...n (in particular its derivative at zero). For finite ensembles the study goes back to the pioneering work of M. Gaudin [14] and later M. Jimbo, T. Miwa, Y. Môri and M. Sato [22]. Forrester and Witte =-=[12]-=-, drawing on [22], have evaluated fβ,n (β = 1, 2) using methods from integrable systems. For example, if β = 1 and n is even:1 f1,n(ε) = τσV (ε 2), where τσV is a function satisfying: σV (t) = t d dt ...