Duminil-Copin–Smirnov theorem

Remarks

Bernard Nienhuis predicted this value in 1982 with the Coulomb gas method. Hugo Duminil-Copin and Stanislav Smirnov proved it in a paper published in 2012.

Idea of the proof: let $x_c = 1/\sqrt{2 + \sqrt 2}$, and let $Z(x) = \sum_{\gamma} x^{\abs{\gamma}}$ be the generating function of self-avoiding walks from a fixed point, where $\abs{\gamma}$ is the length of $\gamma$. The radius of convergence of $Z$ is $1/\mu$, so the claim is that $Z(x) < \infty$ for $x < x_c$ and $Z(x_c) = \infty$. The proof uses a parafermionic observable, a sum over walks in a domain weighted by $x_c^{\abs{\gamma}}$ and by a complex phase given by the winding of $\gamma$. At $x = x_c$ this observable satisfies a discrete form of half of the Cauchy–Riemann equations, around every vertex. Summing these relations over a trapezoidal domain gives a linear identity between walks ending on the different sides, from which both bounds follow.

Example of the statement at work: since $c_n \ge \mu^n$ for all $n$, the number of self-avoiding walks of $n$ steps from a vertex of $\mathbb H$ is at least $(2 + \sqrt 2)^{n/2} \approx 1.848^n$.

Going deeper

For a student

The connective constant is not universal: it depends on the lattice, unlike critical exponents. The hexagonal lattice is special because of a discrete holomorphic structure at $x_c$, the same kind of structure that appears in Smirnov's proof of Cardy's formula. The theorem was a first rigorous confirmation of the Coulomb gas predictions for the self-avoiding walk. The critical exponents of the self-avoiding walk in the plane, such as the growth $c_n \approx \mu^n n^{11/32}$, are still not proved.