Three-dimensional Ising correlation exponent bounds

At least 8 years old · documented by

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,y∈Z3x,y\in\mathbb{Z}^3,

c0∥x−y∥2−ε≤μβc[σxσy]≤c1∥x−y∥1+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.

References

Primary source

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

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.