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