Supercritical self-avoiding walks are space-filling

Abstract

γ δ ), i.e. its geometric behavior when δ goes to 0. The qualitative behavior is expected

Interacting self-avoiding polygons

“…A corresponding result has been obtained in [1] for a single self-avoiding polygon, and indeed our proof heavily uses their methods and results.…”

“…This strategy is implemented in the proof of Proposition 3.4, which uses a multi-valued map principle. We will start by giving an outline of the approach developed in [1] and state Proposition 3.4, from which we deduce the theorem. Then we will prove that the assumptions of Proposition 3.4 hold in the relevant cases.…”

Self-avoiding walk on the complete graph

“…Suppose that eventually y n ≥ n + n r (subcritical case). Then Γ(n, y n ) = y n n e −yn 1 y n − n 1 − y n (y n − n) 2 + O y 2 n (y n − n) 4 .…”

“…There is no notion of "the" critical value on a finite graph, and any value in the critical window will do 3. In fact, some open questions remain concerning the supercritical phase on tori; see[6] 4. This definition differs from the proposal in [12, Definition 1.2], which calls z n subcritical if lim sup χ(n) zn < ∞, supercritical if lim inf χ (n) zn = ∞, and critical if z n /s is subcritical when s > 1 and supercritical when s < 1.…”

Shifted critical threshold in the loop $ \boldsymbol{O(n)} $ model at arbitrarily small $n$

“…We now prove (4). Using the fact that, for any n smaller than a positive value n 0 , min{λ 1 , λ 1 (n)} = λ 1 (n), using (21) and performing a Taylor expansion, we obtain that, for any n ∈ (0, n 0 ), λ c (n) − 1/µ ≥ λ 1 (n) − 1/µ = (1 + λ 4 1 (n) n) a − 1 µ = a µ λ 4 1 (0) n + O(n 2 ) = a µ 5 n + O(n 2 ).…”