Exactly Solved Models

On self-avoiding polygons and walks: the snake method via polygon joining

“…This argument certainly fits the template offered by the proof of Theorem 1.5 (2). We will present the proof by explaining how to modify the statements and arguments used in the preceding derivation.…”

“…for k ∈ R χ+ε s j ,i+2 in the proof of Lemma 11.3(2) (which is needed to prove the third part). That k ∈ R χ+ε s j ,i+2 ensures that both θ k and θ s L −16−k are at most 3/2 + χ + ε = θ, from which follows (14.3), and thus Lemma 11.3 (2) and (3).…”

“…Recall first of all that the hypotheses of Theorem 1.5 (2) implied the existence of the closing exponent, and that we chose to label χ ∈ R such that P n Γ closes = n −1/2−χ+o(1) for odd n; moreover, χ could be supposed to be non-negative. Specifically, it was the lower bound P n Γ closes ≥ n −1/2−χ−o(1) that was invoked, in part one of the preceding proof.…”

“…The set R i may be thought as a set of regular indices j ∈ 2N ∩ 2 i , 2 i+1 , with θ j for example being neither atypically high nor low. As we prove Theorem 1.5 (2), all indices are regarded as regular, so that we make the trivial specification R i = 2N ∩ 2 i , 2 i+1 ; the set will be respecified non-trivially in the proof of Theorem 1.5(1).…”

Solving multivariate functional equations

“…The combinatorial significance of a term like F (sq) is that the statistic marked by q increases at the same time as the statistic marked by s as the objects are being built. Functional equations of this type arise naturally when enumerating with statistics combinatorial objects such as polyominoes [6], plane trees [2], lattice paths [3], and pattern-avoiding permutations [4]. Bousquet-Mélou proved in [5,Lemma 2.3] that the solution to Equation (1.1) is…”