The logarithmic susceptibility asymptotic for the four-dimensional Ising model

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Let χ(β)\chi(\beta) denote the susceptibility of the high-dimensional Ising model and let βc\beta_c be its critical inverse temperature. Logarithmic susceptibility conjecture. Let d=4d=4. There exists A>0A>0 such that

χ(β)=Alog(1β/βc)1/31β/βc(1+o(1)),\chi(\beta)=\frac{A|\log (1-\beta/\beta_c)|^{1/3}}{1-\beta/\beta_c}(1+o(1)),

where o(1)o(1) tends to 00 as β\beta tends to βc\beta_c. Renormalization group analysis and universality suggest this four-dimensional logarithmic correction to mean-field susceptibility behavior; the statement remains open in the source.

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Primary source

Romain Panis, “The incipient infinite cluster of the FK-Ising model in dimensions d3 and the susceptibility of the high-dimensional Ising model”, arXiv:2406.15243 (2025).

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