Three-dimensional Ising correlation exponent bounds

Consider the critical three-dimensional Ising model, with spin variables σx\mathit{\sigma_x} and critical Gibbs measure μβc\mathit{\mu_{\beta_c}}. Three-dimensional Ising exponent conjecture. There exist ε>0\mathit{\varepsilon}>0 and c0,c1(0,)c_0,c_1\in(0,\infty) such that for all x,yZ3x,y\in\mathbb{Z}^3,

c0xy2εμβc[σxσy]c1xy1+ε.\frac{c_0}{\|x-y\|^{2-\varepsilon}}\le \mu_{\beta_c}[\sigma_x\sigma_y]\le \frac{c_1}{\|x-y\|^{1+\varepsilon}}.

The source describes this improvement over the best known three-dimensional bounds as an open problem of significant value.

Sources & referencesView supporting material

Primary source

Hugo Duminil-Copin, “Lectures on the Ising and Potts models on the hypercubic lattice”, arXiv:1707.00520 (2017).

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