The distribution function of a semiflexible polymer and random walks with constraints

In studying the end-to-end distribution function G(r, N) of a worm-like chain by using the propagator method we have established that the combinatorial problem of counting the paths contributing to G(r, N) can be mapped onto the problem of random walks with constraints, which is closely related to the representation theory of the Temperley-Lieb algebra. By using this mapping we derive an exact expression of the Fourier-Laplace transform of the distribution function, G(k, p), as a matrix element of the inverse of an infinite rank matrix. Using this result we also derived a recursion relation permitting to compute G(k, p) directly. We present the results of the computation of G(k, N) and its moments. The moments [r(2n)] of G(r, N) can be calculated exactly by calculating the (1, 1) matrix element of 2n-th power of a truncated matrix of rank n + 1.