Phase transitions for nonsingular Bernoulli actions

The paper’s Theorem C / Theorem 4.2 establishes a sharp phase transition for generalized nonsingular Bernoulli actions on trees: dissipative up to compact stabilizers when 1−H2(µ0,µ1)<e^{-δ/2} and weakly mixing when 1−H2(µ0,µ1)>e^{-δ/2} (statement and setup in Theorem C and Theorem 4.2) . The proof computes the Radon–Nikodym cocycle along the geodesic [ρ,g·ρ] via S_{g·ρ} and shows ∫_X∑_{v∈G·ρ}exp(S_v/2)dµ=∑_{v∈G·ρ}(1−H2)^{2 d(ρ,v)}, matching the candidate’s Hellinger-factorization I(g)=θ^{2 d(ρ,g·ρ)} with θ=1−H2 (paper’s computation and identity d(g·µ)/dµ=exp(S_{g·ρ}) in (4.3)-(proof of Thm 4.2) ). The group integral reduces via the compact-open stabilizer G_ρ to a weighted sum over the orbit G·ρ, and convergence is governed by the Poincaré exponent δ, exactly as argued by the candidate (see the dissipative side of the proof ). For the weak-mixing side, the paper proves infinite recurrence using a Chernoff–Cramér large-deviations estimate and a quasi-Bernoulli percolation on a coarse-grained tree S, then applies the Schmidt–Walters product-ergodicity argument to deduce weak mixing for products with ergodic pmp actions (construction and percolation step ; conclusion and product ergodicity step ). The candidate solution follows the same blueprint and gets the same threshold. Minor issues: the candidate’s large-deviations sign for the ‘good blocks’ is reversed relative to the paper (they write S≤−cM where the paper retains blocks with sum ≥0), but this is a typographical slip that does not affect the intended argument; their dissipativity argument explicitly supplies the inequality ∑a_i^2≤(∑a_i)^2 that the paper uses implicitly when passing from ∑exp(S_v/2) to ∑exp(S_v). Overall, both are correct and essentially the same proof.