387 research outputs found
Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix
We have obtained the exact asymptotics of the determinant . Inverse symbolic
computing methods were used to obtain exact analytical expressions for all
terms up to relative order to the leading term. This determinant is
known to give weighted enumerations of cyclically symmetric plane partitions,
weighted enumerations of certain families of vicious walkers and it has been
conjectured to be proportional to the one point function of the O loop
model on a cylinder of circumference . We apply our result to the loop model
and give exact expressions for the asymptotics of the average of the number of
loops surrounding a point and the fluctuation in this number. For the related
bond percolation model, we give exact expressions for the asymptotics of the
probability that a point is on a cluster that wraps around a cylinder of even
circumference and the probability that a point is on a cluster spanning a
cylinder of odd circumference.Comment: Version accepted by JCTA. Introduction rewritte
Exact conjectured expressions for correlations in the dense O loop model on cylinders
We present conjectured exact expressions for two types of correlations in the
dense O loop model on square lattices with periodic
boundary conditions. These are the probability that a point is surrounded by
loops and the probability that consecutive points on a row are on the
same or on different loops. The dense O loop model is equivalent to the
bond percolation model at the critical point. The former probability can be
interpreted in terms of the bond percolation problem as giving the probability
that a vertex is on a cluster that is surrounded by \floor{m/2} clusters and
\floor{(m+1)/2} dual clusters. The conjectured expression for this
probability involves a binomial determinant that is known to give weighted
enumerations of cyclically symmetric plane partitions and also of certain types
of families of nonintersecting lattice paths. By applying Coulomb gas methods
to the dense O loop model, we obtain new conjectures for the asymptotics
of this binomial determinant.Comment: 17 pages, replaced by version accepted by JSTA
Schur polynomials, banded Toeplitz matrices and Widom's formula
We prove that for arbitrary partitions and integers the sequence of Schur
polynomials for sufficiently large, satisfy a
linear recurrence. The roots of the characteristic equation are given
explicitly. These recurrences are also valid for certain sequences of minors of
banded Toeplitz matrices.
In addition, we show that Widom's determinant formula from 1958 is a special
case of a well-known identity for Schur polynomials
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
We present determinant formulae for the number of tilings of various domains
in relation with Alternating Sign Matrix and Fully Packed Loop enumeration
Convex hulls of random walks, hyperplane arrangements, and Weyl chambers
We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments.
This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of Rn of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement.
We obtain analogous distribution-free results for Weyl chambers of type An−1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form B1 ×···× B1 recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin.
We also give an asymptotic analysis of the obtained absorption probabilities as n → ∞, in both cases of fixed and increasing dimension d
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