A black-box random-walk proof establishes mean-field near-critical decay |x|^{-d+2+ε} exp(-c|x|/ξ) for two-point functions on Z^d (d>2) under a short list of assumptions, covering self-avoiding walk, percolation, Ising, XY, |φ|^4 and lattice trees above their upper critical dimensions.
Hutchcroft, Critical long-range percolation i: High ef- fective dimension (2025), arXiv:2508.18807
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In high-dimensional critical percolation the rescaled k-point connection probability converges to an explicit constant, confirming the Aizenman-Newman conjecture.
A controlled ε-δ expansion around the LR-SR boundary yields two-loop expressions for ν, η_ω and η_k in long-range quantum O(n) models together with a proposed universality diagram.
Adele-valued random walks converge weakly to an adelic Lévy process in the J1 Skorokhod topology after appropriate scaling.
Proves unique phase transition and critical exponents for percolation on hierarchical lattices, with results on the infinite limit graph, noise sensitivity, and a fixed-point condition for monotone Boolean functions.
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A random walk approach to high-dimensional critical phenomena
A black-box random-walk proof establishes mean-field near-critical decay |x|^{-d+2+ε} exp(-c|x|/ξ) for two-point functions on Z^d (d>2) under a short list of assumptions, covering self-avoiding walk, percolation, Ising, XY, |φ|^4 and lattice trees above their upper critical dimensions.
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Convergence of $k$-point functions in high dimensional percolation
In high-dimensional critical percolation the rescaled k-point connection probability converges to an explicit constant, confirming the Aizenman-Newman conjecture.
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Perturbative Renormalization and Universality Diagram for Long-Range Quantum Criticality
A controlled ε-δ expansion around the LR-SR boundary yields two-loop expressions for ν, η_ω and η_k in long-range quantum O(n) models together with a proposed universality diagram.
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A Scaling Limit of Random Walks in the Rational Adeles
Adele-valued random walks converge weakly to an adelic Lévy process in the J1 Skorokhod topology after appropriate scaling.
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Percolation on hierarchical lattices
Proves unique phase transition and critical exponents for percolation on hierarchical lattices, with results on the infinite limit graph, noise sensitivity, and a fixed-point condition for monotone Boolean functions.