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191
J.Propp, The shape of a typical boxed plane partition
 J. of Math
, 1998
"... Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the ..."
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Cited by 57 (5 self)
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Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.
Specification Analysis of Option Pricing Models Based on TimeChanged Lévy Processes
, 2003
"... We analyze the specifications of option pricing models based on timechanged Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. O ..."
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Cited by 43 (7 self)
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We analyze the specifications of option pricing models based on timechanged Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we must incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.
WeylTitchmarsh MFunction Asymptotics, Local Uniqueness Results, Trace Formulas, And BorgType Theorems For Dirac Operators
 Proc. London Math. Soc
, 2001
"... We explicitly determine the highenergy asymptotics for WeylTitchmarsh matrices associated with general Diractype operators on halflines and on R. We also prove new local uniqueness results for Diractype operators in terms of exponentially small differences of WeylTitchmarsh matrices. As con ..."
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Cited by 34 (17 self)
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We explicitly determine the highenergy asymptotics for WeylTitchmarsh matrices associated with general Diractype operators on halflines and on R. We also prove new local uniqueness results for Diractype operators in terms of exponentially small differences of WeylTitchmarsh matrices. As concrete applications of the asymptotic highenergy expansion we derive a trace formula for Dirac operators and use it to prove a Borgtype theorem.
Vicious walkers, friendly walkers and Young tableaux: II With a wall
 J. Phys. A: Math. Gen
"... Research supported by the Australian Research Council. ..."
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Cited by 32 (4 self)
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Research supported by the Australian Research Council.
Analytic Variations on the Airy Distribution
, 2001
"... The Airy distribution (of the “area ” type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain cur ..."
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Cited by 22 (4 self)
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The Airy distribution (of the “area ” type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders −1, −3, −5, etc., as well as + 1 5 11 7 13 19 3, − 3, − 3, etc. and − 3, − 3, − 3, etc. Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on “nonprobabilistic ” arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +∞, and power symmetric functions of the zeros −αk of Ai(z).
A series expansion of fractional Brownian motion with Hurst index exceeding 1/2
, 2002
"... Let B be a fractional Brownian motion with Hurst index H _> 1/2. Denote by Xl < x2 < ... the positive, real zeros of the Bessel function JH of the first kind of orderH, and by yl y2 '" the positive zeros of J1H. We prove the series representation Bt __ sinxnt xn q_ 1csy nt ..."
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Cited by 21 (2 self)
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Let B be a fractional Brownian motion with Hurst index H _> 1/2. Denote by Xl < x2 < ... the positive, real zeros of the Bessel function JH of the first kind of orderH, and by yl y2 '" the positive zeros of J1H. We prove the series representation Bt __ sinxnt xn q_ 1csy nt n=l Xn n=l Yn where X1,X2,... and 1,2,... are independent, Gaussian random variables with mean zero and VarXn  2ctx2IIJ__2i_i(xn), VarYn  2cty2IIJ_t(Yn), where the constant c/ is defined by c  r12HF(2H)sinrH. With probability 1, the random series converges absolutely and uniformly in t C [0,1]. To keep the exposition transparant, we deliberately exclude the case H 1/2. The expansion is still valid in this case, but the proof requires additional technicalities.
AN ANALOGUE OF HARDY’S THEOREM FOR VERY RAPIDLY DECREASING FUNCTIONS ON SEMISIMPLE LIE GROUPS
, 1997
"... We generalise a result of Hardy, which asserts the impossibility of a function and its Fourier transform to be simultaneously “very rapidly decreasing”, to: (i) all noncompact, semisimple Lie groups with one conjugacy class of Cartan subgroups; (ii) SL(2, R); and (iii) all symmetric spaces of the n ..."
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Cited by 18 (2 self)
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We generalise a result of Hardy, which asserts the impossibility of a function and its Fourier transform to be simultaneously “very rapidly decreasing”, to: (i) all noncompact, semisimple Lie groups with one conjugacy class of Cartan subgroups; (ii) SL(2, R); and (iii) all symmetric spaces of the noncompact type. and αβ> 1 4 1. Introduction. A celebrated theorem of L. Schwartz asserts that a function f on R is ‘rapidly decreasing ’ (or in the ‘Schwartz class’) iff its Fourier transform is ‘rapidly decreasing’. Since this theorem is of fundamental importance in harmonic analysis, there is a whole body of literature devoted to generalizing this result
Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators
 Duke Math. J
, 1998
"... Abstract. We prove new criteria of stability of the absolutely continuous spectrum of onedimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturba ..."
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Cited by 17 (6 self)
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Abstract. We prove new criteria of stability of the absolutely continuous spectrum of onedimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all poten2 − tials V (x) satisfying V (x)  ≤ C(1 + x) 3 −ǫ. The main new technique includes an a.e. convergence theorem for a class of integral operators.
Asymptotic laws for regenerative compositions: Gamma subordinators and the like
 PROBAB. THEORY RELATED FIELDS 135
, 2008
"... For ˜ R = 1 −exp(−R) a random closed set obtained by exponential transformation of the closed range R of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of ˜ R. We fo ..."
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Cited by 14 (7 self)
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For ˜ R = 1 −exp(−R) a random closed set obtained by exponential transformation of the closed range R of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of ˜ R. We focus on the number of parts Kn of the composition when ˜ R is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for Kn and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+.
The Average Case Analysis of Algorithms: Mellin Transform Asymptotics
, 1996
"... This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It reviews the use of MellinPerron formulae and of Mellin transforms in this context. Applications include: dividea ..."
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Cited by 12 (0 self)
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This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It reviews the use of MellinPerron formulae and of Mellin transforms in this context. Applications include: divideandconquer recurrences, maxima finding, mergesort, digital trees and plane trees.