137 research outputs found
Combinatorial interpretation and positivity of Kerov's character polynomials
Kerov's polynomials give irreducible character values in term of the free
cumulants of the associated Young diagram. We prove in this article a
positivity result on their coefficients, which extends a conjecture of S.
Kerov. Our method, through decomposition of maps, gives a description of the
coefficients of the k-th Kerov's polynomials using permutations in S(k). We
also obtain explicit formulas or combinatorial interpretations for some
coefficients. In particular, we are able to compute the subdominant term for
character values on any fixed permutation (it was known for cycles).Comment: 33 pages, 13 figures, version 3: minor modifcation
Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
This paper reviews known results which connect Riemann's integral
representations of his zeta function, involving Jacobi's theta function and its
derivatives, to some particular probability laws governing sums of independent
exponential variables. These laws are related to one-dimensional Brownian
motion and to higher dimensional Bessel processes. We present some
characterizations of these probability laws, and some approximations of
Riemann's zeta function which are related to these laws.Comment: LaTeX; 40 pages; review pape
Constructions for cyclic sieving phenomena
We show how to derive new instances of the cyclic sieving phenomenon from old
ones via elementary representation theory. Examples are given involving objects
such as words, parking functions, finite fields, and graphs.Comment: 18 pages, typos fixed, to appear in SIAM J. Discrete Mat
-Martin boundary of killed random walks in the quadrant
We compute the -Martin boundary of two-dimensional small steps random
walks killed at the boundary of the quarter plane. We further provide explicit
expressions for the (generating functions of the) discrete -harmonic
functions. Our approach is uniform in , and shows that there are three
regimes for the Martin boundary.Comment: 18 pages, 2 figures, to appear in S\'eminaire de Probabilit\'e
Confinement, Turbulence and Diffraction Catastrophes
Many features of large N_c transition that occurs in the spectral density of
Wilson loops as a function of loop area (observed recently in numerical
simulations of Yang-Mills theory by Narayanan and Neuberger) can be captured by
a simple Burgers equation used to model turbulence. Spectral shock waves that
precede this asymptotic limit exhibit universal scaling with N_c, with indices
that can be related to Berry indices for diffraction catastrophes.Comment: Presented at PANIC 200
A CLT for Plancherel representations of the infinite-dimensional unitary group
We study asymptotics of traces of (noncommutative) monomials formed by images
of certain elements of the universal enveloping algebra of the
infinite-dimensional unitary group in its Plancherel representations. We prove
that they converge to (commutative) moments of a Gaussian process that can be
viewed as a collection of simply yet nontrivially correlated two-dimensional
Gaussian Free Fields. The limiting process has previously arisen via the global
scaling limit of spectra for submatrices of Wigner Hermitian random matrices.
This note is an announcement, proofs will appear elsewhere.Comment: 12 page
Multiplicative anomaly and zeta factorization
Some aspects of the multiplicative anomaly of zeta determinants are
investigated. A rather simple approach is adopted and, in particular, the
question of zeta function factorization, together with its possible relation
with the multiplicative anomaly issue is discussed. We look primordially into
the zeta functions instead of the determinants themselves, as was done in
previous work. That provides a supplementary view, regarding the appearance of
the multiplicative anomaly. Finally, we briefly discuss determinants of zeta
functions that are not in the pseudodifferential operator framework.Comment: 20 pages, AIP styl
Employee benefits and challenges of telecommuting virtual working arrangements in the services industry
M. Comm.Virtual working arrangements, including telecommuting, are on the increase globally due to the challenges that organisations face in the current global economy. Virtual working arrangements present considerable possible benefits to organisations, employees and the community at large if correctly implemented. It is estimated that 45 million Americans teleworked in 2006 alone (O’Brien & Hayden, 2007) with predictions of the number reaching 100 million in the United States of America by 2010 (Wilsker, 2008). However, in South Africa this organisational form is not well documented or implemented presently. As a result, local organisations are unaware of the employee benefits and challenges that will be faced when implementing a telecommuting programme and how best to implement teleworking arrangements with these factors in mind
System of Complex Brownian Motions Associated with the O'Connell Process
The O'Connell process is a softened version (a geometric lifting with a
parameter ) of the noncolliding Brownian motion such that neighboring
particles can change the order of positions in one dimension within the
characteristic length . This process is not determinantal. Under a special
entrance law, however, Borodin and Corwin gave a Fredholm determinant
expression for the expectation of an observable, which is a softening of an
indicator of a particle position. We rewrite their integral kernel to a form
similar to the correlation kernels of determinantal processes and show, if the
number of particles is , the rank of the matrix of the Fredholm determinant
is . Then we give a representation for the quantity by using an -particle
system of complex Brownian motions (CBMs). The complex function, which gives
the determinantal expression to the weight of CBM paths, is not entire, but in
the combinatorial limit it becomes an entire function providing
conformal martingales and the CBM representation for the noncolliding Brownian
motion is recovered.Comment: v3: AMS_LaTeX, 25 pages, no figure, minor corrections made for
publication in J. Stat. Phy
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