arXiv subjects

Tasmin Chu

Publications and source records attributed to Tasmin Chu.

3 recordsLinked to original sources

arXiv2026

Applications of the cluster graphing

In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proof of this result. We also point out other uses of the cluster graphing construction in the literature, for instance in showing nonuniqueness at $p_u$ in arXiv:1509.00247 [math.GR]. The purpose of this short note is to make folklore proofs known to experts more accessible to the wider community of probabilists and measured group theorists.

math.PR↗

arXiv2025

Heavy repulsion of clusters in Bernoulli percolation

We study Bernoulli$(p)$ percolation on (non)unimodular quasi-transitive graphs and prove that, almost surely, for any two heavy clusters $C$ and $C'$, the set of vertices in $C$ within distance one of $C'$ is light, i.e. it has finite total weight. This is a significant step towards resolving a longstanding question posed by H\"aggstr\"om, Peres, and Schonmann, and a generalization of a theorem of Tim\'ar, who proved the same result in the unimodular setting. Our proof adapts Tim\'ar's approach but requires developing weighted analogues of several classical unimodular results. This presents nontrivial challenges, since in a nonunimodular graph a subtree with infinitely many ends may be hyperfinite or even light. To overcome this, we employ newly developed machinery from the theory of measure-class-preserving equivalence relations and graphs. In particular, we establish a weighted generalization of a theorem of Benjamini, Lyons, and Schramm on the existence of an invariant random subgraph with positive weighted Cheeger constant, a result of independent interest.

math.PR↗

arXiv2024

Oscillating and nonsummable Radon-Nikodym cocycles along the forward geodesic of measure-class-preserving transformations

We consider the least-deletion map on the Cantor space, namely the map that changes the first 1 in a binary sequence to 0, and construct product measures on $2^\mathbb{N}$ so that the corresponding Radon-Nikodym cocycles oscillate or converge to zero nonsummably along the forward geodesic of the map. These examples answer two questions of Tserunyan and Tucker-Drob. We analyze the oscillating example in terms of random walks on $\mathbb{Z}$, using the Chung-Fuchs theorem.

math.DS↗