Paul Rax | Searcharxiv

arXiv2026

The Relay Random Tree: A Stochastic Geometry Approach of Multihop Relay in an Urban Visibility Setting

In a recent work (Lee, Baccelli $'25$), a one dimensional stochastic geometry model was introduced to study Line of Sight (LoS) connections using Reconfigurable Intelligent Surfaces (RIS), in the context of non terrestrial networks. In this model, signal can be propagated in a urban environment, with buildings acting as obstacles with RIS (which, for the scope of this present article can essentially be thought of as relays) on their rooftops, relaying the connection. The present paper extends this model by both allowing arbitrary distributions for the buildings heights, and considering multi-hop connections. Those generalities also lead to considering structural problems linked to the total load of a relay. Furthermore, studying this Line of Sight connection geometry at the light of geometric random graph theory, we show that it constitutes a computationally well understood example that highlights the different classes of the Eternal Family Trees (EFTs) classification.

math.PR↗

arXiv2025

Stochastic domination and lifts of random variables in percolation theory

Consider some matrix waiting for its coefficients to be written. For each column, sample independently a Bernoulli random variable of some parameter $p$. Seeing all this and possibly using extra randomness, Alice then chooses one spot in each column, in any way she wants. When the Bernoulli random variable of some column is equal to 1, the number 1 is written in the chosen spot. When the Bernoulli random variable of a column is 0, nothing is done on this column. We prove that, using extra randomness, it is possible for Bob to fill the empty entries with well chosen 0's and 1's so that the entries of the matrix are independent Bernoulli random variables of parameter $p$. We investigate various generalisations and variations of this problem, and use this result to revisit and generalise (nonstrict) monotonicity of the percolation threshold $p_c$ with respect to a form of graph-quotienting, namely fibrations. We also use this result to revisit the BK inequality. In a second part, which is independent of the first one, we revisit strict monotonicity of $p_c$ with respect to fibrations, a result that naturally requires more assumptions than its nonstrict counterpart. We reprove the bond-percolation case of the result of Martineau--Severo without resorting to essential enhancements, using couplings instead.

math.PR↗