Abstract
In the present article we consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in [9]. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*} $, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.
Highlights
Cable processes constitute a potent tool in conjunction with Dynkin-type isomorphism theorems as shown in the recent articles [9], [10], [12], [21]
In a general enough set-up, we obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed in [9]
We show that for all d ≥ 2, 0 < h∗ < 2u∗, where u∗ stands for the critical value for the percolation of the vacant set of random interlacements on the (d+1)-regular tree, an explicit quantity by the results of [19]
Summary
Cable processes constitute a potent tool in conjunction with Dynkin-type isomorphism theorems as shown in the recent articles [9], [10], [12], [21]. We apply our results to level-set percolation of the Gaussian free field on the (d + 1)-regular tree (d ≥ 2) endowed with unit weights. We prove (0.8) in Corollary 2.5 (see Remark 2.6) It is a direct consequence of a coupling between the Gaussian free field on the cable system and the random. It refines the coupling in [9], in essence, through the use of the strong Markov property of the Gaussian free field on the cable system.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
