Poly(amidoamine)s for Gene Delivery

Random polymers and generalized urn processes

“…Before going further we need to introduce the HLS process [4][5][6][7] and sketch some of its main properties we will use in the following. An HLS urn is a two color urn process that is governed by a functional parameter π (y) called urn function [7]. Let us consider an infinite capacity urn containing two kinds of elements, say black and white balls, and denote by…”

“…that in [7] are shown to be equivalent to the Baghi-Pal model [3,6], a widely investigated model due to its relevance in studying branching phenomena and random trees (see [1][2][3] for some reviews). Linear urn functions satisfy the differential equation…”

“…Although the solution depends on the considered parameter region, for our analysis it will suffice to take a > 0, a + b < 1 and b > 0, β > 0. From Corollary 10 of [7] we have that…”

“…The main point in this method is that it is possible to relate the density of states of an interacting chain with the problem of computing the large deviation behavior of an associated Markov urn process [1][2][3] once the urn function of the problem is identified (see below). Here we deal with models that can be related to a two-color urn, first introduced by Hill, Lane and Sudderth [4][5][6][7], for which a detailed Large deviations theory has been recently developed [7]. In particular, we will provide an explicit example by studying the classic Random Walk range problem (RP, [8][9][10][11][12]), that is, computing the number of different lattice sites visited by a random walk of given lenght.…”

“…for all ω N ∈ L N 1 and all t. This condition ensures that the energy can either increase of one unit or do not increase at all when a monomer is added to the endpoint of ω N , and is an important technical point to connect with the HLS urns as it allows to directly identify an increase in energy followed by one step grows with adding a black ball to the associated urn. It is possible to generalize to include more general transition spectra (multicolor urns) but here we consider the binary cases as the LDP for such urns have been already developed in detail [7].…”