Lattice Effects in the Scaling Limit of the Two-Dimensional Self-Avoiding Walk

Abstract

We consider the two-dimensional self-avoiding walk (SAW) in a simply connected domain that contains the origin. The SAW starts at the origin and ends somewhere on the boundary. The distribution of the endpoint along the boundary is expected to differ from the SLE partition function prediction for this distribution because of lattice effects that persist in the scaling limit. We give a precise conjecture for how to compute this lattice effect correction and support our conjecture with simulations. We also give a precise conjecture for the lattice corrections that persist in the scaling limit of the λ-SAW walk.

Compressed self-avoiding walks, bridges and polygons

“…The extra factor of h −1 comes from the fact that we are specifying the exact height. Using (18), we see that as n → ∞, 1 , and A does not depend on n or u. Similarly, using (18), we see that there exists c 1 , c 2 (independent of u) such that with the same value of λ 1 ,…”

“…Then if D is an open subset of C, and z, w are distinct interior points, we can define µ D (z, w) to be the measure µ(z, w) restricted to curves that stay in D. Equivalently, it is the scaling limit as above of SAWs from nz to nw that stay in nD. This collection of measures satisfies the restriction property: if D ⊂ D and z, w ∈ D, then µ D (z, w) is µ D (z, w) restricted to curves that stay in D. Similarly, if z ∈ ∂D and w ∈ D we can define the measure µ D (z, w) provided that the boundary is sufficiently smooth at z (there are lattice issues involved if the boundary of D is not parallel with a coordinate axis (see [18]), but we will not worry about this here. )…”

The Fixed Irreducible Bridge Ensemble for Self-Avoiding Walks

“…Recently, it has been shown that there are lattice effects which should persist in the scaling limit for general domains D [3]. Therefore, one cannot expect equations (1.9) and (1.10) to provide a full description of the scaling limit for general domains D ⊂ C. However, we will be restricting our attention to curves in the domains H and S, for which there are no lattice effects expected to persist in the scaling limit.…”

“…It has also recently been shown in [2] that the pivot algorithm can be used to simulate self-avoiding walks in the strip S. Taking lattice effects into account (see [3]), it should also be possible to simulate the self-avoiding walk in other domains using the pivot algorithm. Recently, Nathan Clisby has developed a very fast implementation of the pivot algorithm, [1], and that is the algorithm that we use for our simulations.…”

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

“…Another potential way to construct a "measure" associated to LQG with matter central charge c M ∈ (1, 25) is to directly extend the definition of Gaussian multiplicative chaos in the case when c M < 1 (see also Section 2.3). A potential discrete analog of SLE with c M ∈ (1, 25) is the so-called λ-self avoiding walk with λ = −c M /2, which was introduced by Kennedy and Lawler in [KL13]. This model is expected to converge to SLE κ for κ ∈ (0, 4] in the case when λ ≥ −1/2, equivalently, c M ≤ 1, but also makes sense for c M > 1.…”