6 research outputs found
Solving the Frustrated Spherical Model with q-Polynomials
We analyse the Spherical Model with frustration induced by an external gauge
field. In infinite dimensions, this has been recently mapped onto a problem of
q-deformed oscillators, whose real parameter q measures the frustration. We
find the analytic solution of this model by suitably representing the
q-oscillator algebra with q-Hermite polynomials. We also present a related
Matrix Model which possesses the same diagrammatic expansion in the planar
approximation. Its interaction potential is oscillating at infinity with period
log(q), and may lead to interesting metastability phenomena beyond the planar
approximation. The Spherical Model is similarly q-periodic, but does not
exhibit such phenomena: actually its low-temperature phase is not glassy and
depends smoothly on q.Comment: Latex, 14 pages, 2 eps figure
Area versus Length Distribution for Closed Random Walks
Using a connection between the -oscillator algebra and the coefficients of
the high temperature expansion of the frustrated Gaussian spin model, we derive
an exact formula for the number of closed random walks of given length and
area, on a hypercubic lattice, in the limit of infinite number of dimensions.
The formula is investigated in detail, and asymptotic behaviours are evaluated.
The area distribution in the limit of long loops is computed. As a byproduct,
we obtain also an infinite set of new, nontrivial identities.Comment: 17 page