15:30 - 16:30, September 10, 2024 (Tuesday) Room 440, Astronomy-Mathematics Building, NTU + Cisco WebEx, Physical+Online Seminar
(實體+線上演講 台灣大學天文數學館440室 + Cisco WebEx)
Attempts to Prove a Local Limit Theorem for the Long-range Self-avoiding Walk Yoshinori Kamijima (Toyo University)

Abstract

The self-avoiding walk (SAW) is a model defined by adding self-avoidance interaction to the random walk. In other words, each path does not visit the same vertex on a graph more than once. We consider the connectivity function defined by the number of -step SAWs from the origin to a vertex . It is known that the spread-out SAW with finite-range interactions enjoys the central limit theorem [van der Hofstad and Slade (2002) PTRF][van der Hofstad and Slade (2003) AAM]. Taking an average on a ball, they also proved a certain type of a local limit theorem for . For the spread-out SAW with long-range interactions whose one-step distribution has heavy tails, the power-law decay of the two-point function was shown in [Chen and Sakai (2015) AOP][Chen and Sakai (2019) CMP].

In this talk, I will explain an attempt to prove a local limit theorem for the spread-out long-range SAW in the original sense. Our motivations come from combining the results of the previous researches. I will show two different strategies. The lace expansion gives a certain type of a recurrence relation for the sequence . The first one is based on the analogous approach with [Chen and Sakai (2019) CMP] in which we substitute the recurrence relation into . The second one is based on the inductive approach [van der Hofstad and Slade (2002) PTRF] extended to the long-range model in which we assume an upper bound on for and prove it for . I will report the current progress of our attempts using these approaches.

This talk is joint work with Lung-Chi Chen (National Chengchi University) and Yuki Chino (National Yang-Ming Chiao-Tung University).