Abstract
Following similar analysis to that in Lacoin (Probab Theory Relat Fields 159: 777–808, 2014), we can show that the quenched critical point for self-avoiding walk on random conductors on \(\mathbb {Z}^d\) is almost surely a constant, which does not depend on the location of the reference point. We provide upper and lower bounds which are valid for all \(d\ge 1\).
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Notes
One of two anonymous referees found the following much simpler proof of (2.12). First, by the trivial inequality \(|{\hat{\Omega }}^{\scriptscriptstyle \mathsf {good}}_{\delta ,\varvec{X}}(x;n)|\le c(n)\), we obtain
$$\begin{aligned} \mathbb {E}\big [|{\hat{\Omega }}^{\scriptscriptstyle \mathsf {good}}_{\delta ,\varvec{X}}(x;n)|\big ]&\le \frac{1}{2}c(n)\,\mathbb {P}\big (|{\hat{\Omega }}^{\scriptscriptstyle \mathsf {good}}_{\delta ,\varvec{X}}(x;n)|<\tfrac{1}{2}c(n)\big ) +c(n)\,\mathbb {P}\big (|{\hat{\Omega }}^{\scriptscriptstyle \mathsf {good}}_{\delta ,\varvec{X}}(x;n)|\ge \tfrac{1}{2}c(n)\big )\nonumber \\&=\frac{1}{2}c(n)\Big (1+\mathbb {P}\big (|{\hat{\Omega }}^{\scriptscriptstyle \mathsf {good}}_{\delta ,\varvec{X}}(x;n)|\ge \tfrac{1}{2}c(n)\big )\Big ). \end{aligned}$$(2.11)Combining this with (2.10), we can readily conclude \(\mathbb {P}\big (|{\hat{\Omega }}^{\scriptscriptstyle \mathsf {good}}_{\delta ,\varvec{X}}(x;n)|\ge \tfrac{1}{2}c(n)\big )\ge 1-o(1)\).
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Acknowledgments
The authors are deeply indebted to two anonymous referees for their constructive comments and numerous suggestions to improve presentation. We would also like to thank Rongfeng Sun for many valuable suggestions, Hubert Lacoin for clarifying some of the details in his paper [16] and Hugo Duminil-Copin for pointing out typos in a previous version of the manuscript. The first-named author gave a talk at the IMS workshop held in Singapore during May 4–15, 2015, and received inspiring feedback from participants. Finally we are grateful to Satoshi Handa and Dai Kawahara for their continual involvement in this project.
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Chino, Y., Sakai, A. The Quenched Critical Point for Self-Avoiding Walk on Random Conductors. J Stat Phys 163, 754–764 (2016). https://doi.org/10.1007/s10955-016-1477-0
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DOI: https://doi.org/10.1007/s10955-016-1477-0