Conformal invariance of the critical self-avoiding walk on the hexagonal lattice: convergence to chordal SLE(8/3)

Duminil-Copin and Smirnov's 2012 proof that the connective constant of the hexagonal lattice equals sqrt(2+sqrt 2) settled Nienhuis's 1982 prediction, but the deeper predictions about the two-dimensional self-avoiding walk (SAW) recorded in the same paper remain open. The paper conjectures that the critical SAW curve converges, under mesh refinement, to the chordal Schramm-Loewner Evolution with parameter 8/3 (Conjecture 1), and that the normalized parafermionic observable — the winding-weighted partition function that satisfies half of the discrete Cauchy-Riemann relations and drove the connective-constant proof — converges to a conformally covariant limit (Conjecture 2), which would be the main step toward Conjecture 1 and toward the predicted critical exponents gamma = 43/32 and nu = 3/4 and the T^{-1/4} decay of the critical bridge partition function.

Background

A self-avoiding walk (SAW) is a lattice path that visits no vertex more than once; it was introduced in the 1950s as a model of polymer chains and became one of the most studied models of statistical mechanics. On the hexagonal (honeycomb) lattice, the number $c_n$ of $n$-step SAWs grows exponentially and $c_n^{1/n}$ tends to a limit $\mu$, called the connective constant of the lattice. Using the Coulomb-gas formalism for the two-dimensional $O(n)$ model, Nienhuis (1982) predicted $\mu = \sqrt{2+\sqrt2}$; Duminil-Copin and Smirnov (2012) proved this. Their tool was a parafermionic observable: a walk-counting partition function $F(a,z) = \sum_{\gamma: a \to z} e^{-i\sigma W_\gamma(a,z)} x^{\ell(\gamma)}$, where $W_\gamma(a,z)$ is the winding (the total rotation, in radians, along the walk from $a$ to $z$), $\ell(\gamma)$ the walk's length, and $\sigma = 5/8$ a spin parameter. At the critical weight $x_c = 1/\sqrt{2+\sqrt2}$ this observable satisfies a vanishing discrete contour integral around every lattice vertex — one half of the discrete Cauchy–Riemann relations of discrete complex analysis — which was enough to pin down $\mu$ but not to make the observable discrete-holomorphic, since in a domain with $E$ mid-edges the relation provides only about $2E/3$ constraints for $E$ values, leaving the observable, in the authors' words, a divergence-free vector field with conjecturally non-trivial discrete curl.

Section 4 of that paper records the central remaining problems. Lawler, Schramm and Werner (2004) conjectured that the critical SAW has a conformally invariant scaling limit: for a simply connected planar domain $\Omega \ne \mathbb{C}$ with two distinct boundary points $a, b$, the random curve $\gamma_\delta$ drawn from self-avoiding trajectories in the mesh-$\delta$ approximation $(\Omega_\delta, a_\delta, b_\delta)$, weighted proportionally to $x_c^{\ell(\gamma)}$, should converge in law as $\delta \to 0$ to chordal SLE(8/3) — the member of the one-parameter Schramm–Loewner Evolution family of random conformally invariant curves that satisfies the restriction property (invariance under conditioning the curve to remain in subdomains), the continuum analogue of the elementary fact that a SAW conditioned to stay in a subgraph is a SAW on that subgraph. The same physics predicts the critical exponents: $c_n \sim A n^{\gamma-1} \mu^n$ with $\gamma = 43/32$ (Nienhuis), mean-square displacement $\langle|\gamma(n)|^2\rangle = n^{2\nu+o(1)}$ with $\nu = 3/4$ (Flory-type prediction supported by Nienhuis), and, for the partition function $B_T^{x_c}$ of bridges (SAWs crossing a strip of width $T$) at criticality, the decay $B_T^{x_c} \sim T^{-1/4}$. Lawler, Schramm and Werner showed that $\gamma$ and $\nu$ would follow from a conformally invariant scaling limit of the SAW. The analogue of the whole program was completed for the Ising model, where the fermionic observable ($\sigma = 1/2$, corresponding to $n = 1$) satisfies the full set of discrete Cauchy–Riemann relations and converges to a conformally covariant limit (Chelkak and Smirnov 2012); for the SAW ($n = 0$) only half of the relations is available, which is the main structural obstacle recorded in the source. The source expects the missing half of the discrete Cauchy–Riemann relations to hold in the scaling limit — the discrete curl of the observable vanishing, equivalently $F_\delta(z)$ having the same limit regardless of the orientation of the edge $z$ — which, together with the prescribed boundary values, would determine the limit.

Problem Statement

Prove the conformal invariance of the critical self-avoiding walk on the hexagonal lattice, in the formulation of Duminil-Copin and Smirnov (2012), with critical weight $x_c = 1/\sqrt{2+\sqrt2}$:

(1) Conjecture 2. For every simply connected domain $\Omega \ne \mathbb{C}$, every point $z \in \Omega$, and distinct boundary points $a, b$ with $\partial\Omega$ smooth near $b$: let $F_\delta$ be the winding-based parafermionic observable with spin $\sigma = 5/8$ in the mesh-$\delta$ domain $(\Omega_\delta, a_\delta)$, and let $z_\delta, b_\delta$ be the mesh-$\delta$ points closest to $z, b$. Then

$$\lim_{\delta \to 0} \frac{F_\delta(z_\delta)}{F_\delta(b_\delta)} = \left(\frac{\phi'(z)}{\phi'(b)}\right)^{5/8}$$

where $\phi$ is a conformal map from $\Omega$ to the upper half-plane sending $a$ to $\infty$ and $b$ to $0$ (the right-hand side is well defined because $\phi$ is unique up to multiplication by a real factor).

(2) Conjecture 1 (the target). For every simply connected domain $\Omega \ne \mathbb{C}$ with two distinct boundary points $a, b$, the law of the random curve $\gamma_\delta$ in $(\Omega_\delta, a_\delta, b_\delta)$, obtained by weighting each self-avoiding trajectory proportionally to $x_c^{\ell(\gamma)}$, converges as $\delta \to 0$ to the chordal Schramm–Loewner Evolution with parameter $\kappa = 8/3$ in $\Omega$ from $a$ to $b$.

(3) Linked consequences of the same scaling limit. Derive the predicted critical behavior: the asymptotic $c_n \sim A n^{\gamma-1}\mu^n$ with $\mu = \sqrt{2+\sqrt2}$ and $\gamma = 43/32$; the mean-square displacement $\langle|\gamma(n)|^2\rangle = n^{2\nu+o(1)}$ with $\nu = 3/4$; and the decay $B_T^{x_c} \sim T^{-1/4}$ of the critical bridge partition function as $T \to \infty$.

These three parts serve a single scientific objective, the rigorous continuum description of the critical self-avoiding walk: Lawler, Schramm and Werner showed that the exponents in (3) are computable once the scaling limit in (2) is conformally invariant, and the source states that proving (1) 'would be a major step toward' (2) 'and the derivation of critical exponents'. Determination of the square-lattice connective constant and the critical points of the loop $O(n)$ model for $n \in (0, 2]$ are neighboring open problems and lie outside this question.

References

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