Probability studies random structures and their limits.
4 records · Includes focused interests · How we rate progress
OpenAI claims disappearance of infinite bond-percolation clusters at criticality on every infinite connected locally finite quasi-transitive graph whose critical threshold is below one.
Read original source Pinned primary manuscript · family 213github.com
Original report: 2026-10-06Parisi and Zamponi give an analytic proof of a+b=1 from the scaling full replica-symmetry-breaking equations for infinite-dimensional hard-sphere jamming. Together with the earlier relation b=(1+c)/2, it yields the scaling identities linking force, gap and overlap exponents.
Read original source Parisi–Zamponi: precise identity, assumptions and AI disclosurearxiv.org
Original report: 2026-06-02The pinned Anthropic development reports θ(p_c)=0 for nearest-neighbour Bernoulli bond percolation on Zᵈ in every integer dimension d≥2. It proves Kozma–Nitzan’s Conjecture 3 through an additive gluing inequality and formalizes the reduction to the lattice statement.
Read original source Pinned Anthropic development: exact lattice statementgithub.com
Original report: 2026-08-28Christopher D. Long reports explicit complex polynomials P and Q in three independent real Gaussian variables with E(P^m)=0 and E(QP^m)=m! for every positive integer m. A five-term degree-four example refutes the conjecture in every dimension at least three; a six-term cubic example works in four variables.
Read original source Original paper and AI-provenance sectionarxiv.org
Original report: 2026-07-20Probability studies random structures and their limits. Critical thresholds and phase transitions depend on the graph, measure, and model.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
OpenAI reports: Critical percolation on quasi-transitive graphs
Parisi–Zamponi–Claude: an analytic proof of the jamming exponent identity
Claude reports: no infinite critical cluster on integer lattices
AI-assisted small counterexamples to the Gaussian Moments Conjecture
OpenAI reports: Critical percolation on quasi-transitive graphs
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