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Brownian Bridge Asymptotics for Random pMappings
 Electonic J. Probab
, 2002
"... The Joyal bijection between doublyrooted trees and mappings can be lifted to a transformation on function space which takes treewalks to mappingwalks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 ..."
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Cited by 13 (8 self)
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The Joyal bijection between doublyrooted trees and mappings can be lifted to a transformation on function space which takes treewalks to mappingwalks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 AldousPitman result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random pmappings.
Invariance principles for nonuniform random mappings and trees
 ASYMPTOTIC COMBINATORICS WITH APPLICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases ..."
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Cited by 11 (9 self)
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In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases, in this paper we formulate a sampling invariance principle in terms of iterates of a fixed number of random elements. We show that the sampling invariance principle implies many, but not all, of the distributional limits implied by the functional invariance principle. We give direct verifications of the sampling invariance principle in two successive generalizations of the uniform case, to pmappings (where elements are mapped to i.i.d. nonuniform elements) and Pmappings (where elements are mapped according to a Markov matrix). We compare with parallel results in the simpler setting of random trees.
The Asymptotic Distribution of the Diameter of a Random Mapping
, 2002
"... The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an nelement set to itself is represented as the distribution of a functional of a reflecting Brownian bridge. This yields a formula for the Mellin transform of the asymptotic distribution, ge ..."
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Cited by 5 (3 self)
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The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an nelement set to itself is represented as the distribution of a functional of a reflecting Brownian bridge. This yields a formula for the Mellin transform of the asymptotic distribution, generalizing the evaluation of its mean by Flajolet and Odlyzko (1990).
Weak convergence of random pmappings and the exploration process of inhomogeneous continuum random trees
 PROBAB. THEORY RELAT. FIELDS
, 2005
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The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity
, 2004
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Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model)
, 2009
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Forest volume decompositions and AbelCayleyHurwitz multinomial expansions
, 2001
"... This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes define ..."
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Cited by 2 (0 self)
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This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating outdegrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1