[Paper Review] On the scaling limit of planar self-avoiding walk

Gregory F. Lawler, Oded Schramm+1 more|ArXiv.org|Apr 23, 2002

Diffusion and Search Dynamics4 citations

TL;DR

This paper proposes that the scaling limit of planar self-avoiding walks (SAWs) is the stochastic Loewner evolution with parameter $\kappa = 8/3$ ($SLE_{8/3}$), based on conformal invariance and recent advances in SLE theory. It derives critical exponents $\nu = 3/4$ and $\gamma = 43/32$ nonrigorously via SLE calculations, offering a precise conjectural framework for the continuum limit of SAWs and self-avoiding polygons as outer boundaries of Brownian loops.

ABSTRACT

A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the scaling limit of the uniform measures on these objects. The conjectures are based on recent results on the stochastic Loewner evolution and non-disconnecting Brownian motions. New heuristic derivations are given for the critical exponents for SAWs and SAPs.

Motivation & Objective

• To formulate precise conjectures for the scaling limit of uniform measures on planar self-avoiding walks (SAWs) and self-avoiding polygons (SAPs).
• To establish a connection between the scaling limit of SAWs and the stochastic Loewner evolution $SLE_{8/3}$, assuming conformal invariance or covariance.
• To re-derive the critical exponents $\nu = 3/4$ and $\gamma = 43/32$ using rigorous SLE calculations, providing nonrigorous but precise predictions.
• To extend the conjecture that the scaling limit of SAPs corresponds to the outer boundary of Brownian loops, consistent with Mandelbrot’s earlier heuristic.

Proposed method

• Use of stochastic Loewner evolution ($SLE_{\kappa}$) as a candidate scaling limit for SAWs, with $\kappa = 8/3$ proposed based on conformal invariance and locality properties.
• Application of rigorous results on $SLE_{8/3}$ to compute critical exponents, such as $\nu$ and $\gamma$, via its conformal invariance and fractal properties.
• Adaptation of Kesten’s renewal theory and bridge decomposition techniques to prove the existence of the infinite half-space SAW in all dimensions $d$, using the connective constant $\beta$.
• Derivation of the weak limit of uniform measures on $n$-step SAWs in the half-space $\mathbb{Z}^d_+$ as a mixture of i.i.d. irreducible bridges with weights $\beta^{-k}$.
• Use of subadditivity and asymptotic analysis to establish $\lim_{n\to\infty} \upsilon_{n+1}/\upsilon_n = \beta$, where $\upsilon_n$ is the number of $n$-step SAWs in the half-space.
• Formulation of conjectures based on conformal covariance, linking SAW and SAP scaling limits to $SLE_{8/3}$ and outer boundaries of Brownian motion.

Experimental results

Research questions

• RQ1 What is the precise form of the scaling limit of uniform measures on planar self-avoiding walks, assuming conformal invariance or covariance?
• RQ2 Why should the scaling limit of SAWs be $SLE_{8/3}$, and how does this relate to the critical exponent $\nu = 3/4$?
• RQ3 Can the critical exponent $\gamma = 43/32$ for the number of SAWs be re-derived via $SLE_{8/3}$ calculations?
• RQ4 Is the scaling limit of self-avoiding polygons (SAPs) equivalent to the outer boundary of Brownian loops, as conjectured by Mandelbrot?
• RQ5 What is the rigorous status of the existence of the infinite half-space SAW in $\mathbb{Z}^d$ for general $d$?

Key findings

• The scaling limit of planar SAWs is conjectured to be $SLE_{8/3}$ under the assumption of conformal invariance or covariance.
• The critical exponent $\nu = 3/4$ is re-derived via rigorous calculations on $SLE_{8/3}$, consistent with Flory’s and renormalization group predictions.
• The critical exponent $\gamma = 43/32$ is re-derived nonrigorously using $SLE_{8/3}$, confirming Nienhuis’ prediction via conformal field theory.
• The existence of the infinite half-space SAW is rigorously established in all dimensions $d$ via a renewal process and the connective constant $\beta$, showing weak convergence to a mixture of i.i.d. irreducible bridges.
• The limit measure on infinite half-space SAWs is shown to be equivalent to the weak limit of uniform measures on $n$-step bridges, with the limit characterized by weights $\beta^{-k}$ on irreducible bridges of length $k$.
• The conjecture that SAPs scale to the outer boundary of Brownian loops is formalized, with $SLE_{8/3}$ providing a natural candidate for the continuum limit of SAWs.