An exact solution of three interacting friendly walks in the bulk

Abstract

Abstract. We find the exact solution of three interacting friendly directed walks on the square lattice in the bulk, modelling a system of homopolymers that can undergo gelation by introducing two distinct interaction parameters that differentiate between the zipping of only two or all three walks. We establish functional equations for the model's corresponding generating function that are subsequently solved exactly by means of the obstinate kernel method. We then proceed to analyse our model, first considering the case where triple-walk interaction effects are ignored, finding that our model exhibits two phases which we classify as free and gelated regions, with the system exhibiting a second-order phase transition. We then analyse the full model where both interaction parameters are incorporated, presenting the full phase diagram and highlighting the additional existence of a first-order gelation boundary.

Bubble-bound state of triple-stranded DNA: Efimov physics in DNA with repulsion

“…The phase diagrams obtained in Ref. [43] resemble partly the phase diagrams reported here, but without any mixed phase. In this paper we focus on the mixed phase which occurs on the bound side of the two chain melting.…”

“…The Bound phase now melts via the Efimov state for larger X, but at y = yc for smaller X. This phase diagram is similar to that in Ref [43]…”

“…A model of three noncrossing chains in 1 + 1 dimensions, without any cooperativity factor but with a three chain interaction, was solved exactly in Ref. [43]. The phase diagrams obtained in Ref.…”

Exact Solution of Some Quarter Plane Walks with Interacting Boundaries

“…The solutions to these models are varied. Model 18 is very similar to the model considered in [25]; the main difference being the presence of a weighted neutral step. The solution we give in Section 7 is essentially unchanged from that in [25].…”

“…In the statistical physics and probability literatures this type of extension is described as adding an interaction on the boundary. Interaction problems are important as they often lead to so-called phase transitions [13] where the behaviour of the system changes markedly as the variable corresponding to surface visits is varied [24,25]. Here our interest is in how counting the visits to the boundary changes the solvability of the problem and the analytic character of the generating function.…”