2011/05/03 by R. Brak, Richard Brak, Gary K Iliev +3 · 6 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic number #Algebraic structures and combinatorial models #Bijection #Degree (music) #Discriminant #Generating function #Path (computing) #Polynomial #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.CO #math.MP
paper · pdf · doi:10.1007/s10955-011-0306-8
published in Journal of Statistical Physics 145(3), 669-685 (Springer Science+Business Media) · typo in author's name corrected, no further change
arxiv created 2011/05/03 · openalex publication_date 2011/09/08 · arxiv updated 2015/05/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We define (k,ℓ)-restricted Lukasiewicz paths, k≤ℓ∈ℕ0, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of (k,ℓ). The resulting polynomial is of degree ℓ+1 and hence cannot be solved explicitly for sufficiently large ℓ. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterization of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient. We then give a bijection between (k,ℓ)-restricted Lukasiewicz paths and "rise"-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For (k,ℓ)=(1,∞) we provide a new bijection to Motzkin paths. We also consider the area-weighted generating function and show that it is a q-deformed algebraic function. We determine the generating function explicitly in particular cases of (k,ℓ)-restricted Lukasiewicz paths, and for (k,ℓ)=(0,∞) we provide a bijection to Dyck paths.