2001/08/10 by Irene Hueter, Hueter, Irene · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #60G50 (Primary) 60D05 #60G57 (Secondary) #Computational Geometry and Mesh Generation #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematics and Applications #Point processes and geometric inequalities #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60D05 #msc:60G50 #msc:60G57
paper · pdf · doi:10.48550/arxiv.math/0108077
32 pages Change to first version: The lower bound for the normalized first two moments of the distance of the weakly SAW from its starting point is not uniform in βas βtends to infinity
openalex publication_date 2001/08/10 · arxiv created 2001/08/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ν= 3/4. This value is an instance of the formula ν=1 on Z and ν= max(1/2, 1/4 + 1/d) in Zd for dimensions d ≥ 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 Z2 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 β>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.