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Lower order terms in Szego type asymptotic formulas, and combinatorial identities describing the maximum of a random walk

Mathematical Physics & Probability

Speaker: D. Gioev, University of Pennsylvania
Location: 693 Kerr
Start time: Thu, May 9 2002, 3:10PM

The Strong Szego Limit Theorem (SSLT) is a second order asymptotic formula for the determinant of a large Toeplitz matrix. We obtain a third order generalization of SSLT for a pseudodifferential operator on the unit circle and, more generally, on a Zoll manifold of any dimension. A particular case is a Szego type asymptotics for an operator of multiplication by a smooth function on the standard sphere of any dimension. This is a refinement of a result by V.Guillemin and K.Okikiolu who have established a second order generalization. The proof uses the method of Guillemin and Okikiolu and proceeds in the spirit of the combinatorial proof of the classical SSLT by M.Kac. An important role in the proof is played by a certain combinatorial identity which generalizes the formula of G.A.Hunt and F.J.Dyson to an arbitrary natural power. The original Hunt--Dyson combinatorial formula, for the power one, has been used by M.Kac in the mentioned proof of the classical SSLT, and also in a computation of the expected value of the maximum of a random walk with independent identically distributed (i.i.d.) steps. It turns out that the generalized Hunt--Dyson formula is another form of a combinatorial theorem by H.F.Bohnenblust which allowed F.Spitzer to compute the characteristic function of the maximum of a random walk with i.i.d. steps.