1,640 research outputs found
Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
We develop a new method for studying the asymptotics of symmetric polynomials
of representation-theoretic origin as the number of variables tends to
infinity. Several applications of our method are presented: We prove a number
of theorems concerning characters of infinite-dimensional unitary group and
their -deformations. We study the behavior of uniformly random lozenge
tilings of large polygonal domains and find the GUE-eigenvalues distribution in
the limit. We also investigate similar behavior for alternating sign matrices
(equivalently, six-vertex model with domain wall boundary conditions). Finally,
we compute the asymptotic expansion of certain observables in dense
loop model.Comment: Published at http://dx.doi.org/10.1214/14-AOP955 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A note on the probability of generating alternating or symmetric groups
We improve on recent estimates for the probability of generating the
alternating and symmetric groups and . In
particular we find the sharp lower bound, if the probability is given by a
quadratic in . This leads to improved bounds on the largest number
such that a direct product of copies
of can be generated by two elements
A note on the probability of generating alternating or symmetric groups
We improve on recent estimates for the probability of generating the
alternating and symmetric groups and . In
particular we find the sharp lower bound, if the probability is given by a
quadratic in . This leads to improved bounds on the largest number
such that a direct product of copies
of can be generated by two elements
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks
AbstractFuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular, we show that, in some sense, almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman's conjecture that every Fuchsian group surjects onto all large enough alternating groups. As a very special case, we obtain a random Hurwitz generation of An, namely random generation by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higman's conjecture for symmetric groups. We apply these results to branched coverings of Riemann surfaces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Γ has (n!)μ+o(1) index n subgroups, where μ is the measure of Γ, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces. A final application concerns random walks on alternating and symmetric groups. We give necessary and sufficient conditions for a collection of ‘almost homogeneous’ conjugacy classes in An to have product equal to An almost uniformly pointwise. Our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups
Asymptotics for incidence matrix classes
We define {\em incidence matrices} to be zero-one matrices with no zero rows
or columns. A classification of incidence matrices is considered for which
conditions of symmetry by transposition, having no repeated rows/columns, or
identification by permutation of rows/columns are imposed. We find asymptotics
and relationships for the number of matrices with ones in these classes as
.Comment: updated and slightly expanded versio
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