Proof of the Conjecture that the Planar Self-Avoiding Walk has Root Mean Square Displacement Exponent 3/4

TL;DR

This paper proves the long-standing conjecture that the planar self-avoiding walk has a root mean square displacement exponent of 3/4, confirming a key prediction in chemical physics and advancing understanding of its geometric properties.

Contribution

It provides a rigorous proof of the displacement exponent for the planar self-avoiding walk, using point process analysis and Palm distribution techniques.

Findings

Proves the root mean square displacement exponent =3/4 for the 2D self-avoiding walk.

Establishes a framework connecting self-intersection point processes to displacement exponents.

Derives asymptotic expected distance results for weakly self-avoiding walks.

Abstract

This paper proves the long-standing open conjecture rooted in chemical physics (Flory (1949)) that the self-avoiding walk (SAW) in the square lattice has root mean square displacement exponent \nu= 3/4. This value is an instance of the formula \nu=1 on Z and \nu = max(1/2, 1/4 + 1/d) in Z^d for dimensions d \geq 2, which will be proved in a subsequent paper. This expression differs from the one that Flory's arguments suggested. We consider (a) the point process of self-intersections defined via certain paths of the symmetric simple random walk in Z^2 and (b) a ``weakly self-avoiding cone process'' relative to this point process when in a certain "shape". We derive results on the asymptotic expected distance of the weakly SAW with parameter \beta>0 from its starting point, from which a number of distance exponents are immediately collectable for the SAW as well. Our method employs the Palm distribution of the point process of self-intersection points in a cone.