Exact Solution of Semi-Flexible and Super-Flexible Interacting Partially Directed Walks

We provide the exact generating function for semi-flexible and super-flexible interacting partially directed walks and also analyse the solution in detail. We demonstrate that while fully flexible walks have a collapse transition that is second order and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order. This confirms a recent conjecture based on numerical results. We note that the addition of an horizontal force in either case does not affect the order of the transition. In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the super-flexible case, the transition also changes, though more subtly, with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing

A double bounded key identity for Goellnitz's (big) partition theorem

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first time a bounded version of Goellnitz's (big) theorem has been proved. This leads to new bounded versions of Jacobi's triple product identity for theta functions and other fundamental identities.Comment: 17 pages, to appear in Proceedings of Gainesville 1999 Conference on Symbolic Computation

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

All the colours of the rainbow.

Our perception of colour has always been a source of fascination, so it's little wonder that studies of the phenomenon date back hundreds of years. What, though, can modern scientists learn from medieval literature — and how do we go about it

Area versus Length Distribution for Closed Random Walks

Using a connection between the $q$-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

Two parameter Deformed Multimode Oscillators and q-Symmetric States

Two types of the coherent states for two parameter deformed multimode oscillator system are investigated. Moreover, two parameter deformed $gl(n)$ algebra and deformed symmetric states are constructed.Comment: LaTeX v1.2, 14 pages with no figure

Exact Solution of the Discrete (1+1)-dimensional RSOS Model with Field and Surface Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions ${}_3\phi_2$, it introduces a new form of solution to the lexicon of directed lattice path generating functions.Comment: 10 pages, 2 figure

h analogue of Newton's binomial formula

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.Comment: 6 pages, latex Jounal-ref: J. Phys. A: Math. Gen. 31 (1998) L75

Impact of localization on Dyson's circular ensemble

A wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by Dyson's circular ensemble. We argue that the impact of localization in such systems is to provide certain restrictions on the eigenvalues. We consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum, and there is a transition from the universal Wigner distribution towards a Poisson distribution with increasing localization.Comment: To be published in J. Phys.

More on the q-oscillator algebra and q-orthogonal polynomials

Properties of certain $q$-orthogonal polynomials are connected to the $q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to arise as matrix elements of $q$-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous $q$-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big $q$-Hermite polynomials and the continuous $q$-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are algebraically derived