4 research outputs found
A Fast Algorithm for the Inversion of Quasiseparable Vandermonde-like Matrices
The results on Vandermonde-like matrices were introduced as a generalization
of polynomial Vandermonde matrices, and the displacement structure of these
matrices was used to derive an inversion formula. In this paper we first
present a fast Gaussian elimination algorithm for the polynomial
Vandermonde-like matrices. Later we use the said algorithm to derive fast
inversion algorithms for quasiseparable, semiseparable and well-free
Vandermonde-like matrices having complexity. To do so we
identify structures of displacement operators in terms of generators and the
recurrence relations(2-term and 3-term) between the columns of the basis
transformation matrices for quasiseparable, semiseparable and well-free
polynomials. Finally we present an algorithm to compute the
inversion of quasiseparable Vandermonde-like matrices
Heat kernels on 2d Liouville quantum gravity: a numerical study
We numerically compute the heat kernel on a square lattice torus equipped
with the measure corresponding to Liouville quantum gravity (LQG). From the
on-diagonal heat kernel we verify that the spectral dimension of LQG is 2.
Furthermore, when diffusion is started from a high point of the underlying
Gaussian free field, our numerics indicates superdiffusive space-time scaling
with respect to the Euclidean metric in the small space-to-time regime. The
implications of this result require further investigation, but seem to coincide
with the notion that the Euclidean metric is not the right geodesic for
characterizing the geometry of LQG.Comment: Preliminary repor