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

The rigorous computation by Duminil-Copin and Smirnov of the connective constant of the hexagonal lattice, $\mu=\sqrt{2+\sqrt 2}$, left the central structural question of the two-dimensional self-avoiding walk open: prove that the critical walk admits a conformally invariant scaling limit — specifically, that the random self-avoiding curve in a discrete domain converges to chordal Schramm–Loewner evolution with parameter $\kappa=8/3$ (their Conjecture 1). Such a limit is expected to deliver the long-standing Nienhuis predictions: $c_n\sim A n^{\gamma-1}\mu^n$ with $\gamma=43/32$, mean-square displacement $n^{2\nu}$ with $\nu=3/4$, and critical bridge decay $T^{-1/4}$. All of these remain open: the only rigorous SLE(8/3) convergence theorems for self-avoiding walk concern random quadrangulations, and rigorous progress toward the exponents stops at qualitative and quantitative sub-ballisticity plus polynomial (exponent-free) bridge decay.

Background

A self-avoiding walk (SAW) is a nearest-neighbor path on a lattice that visits no vertex more than once; proposed in the 1950s as a model of polymer chains, it has become a fundamental and famously intractable object of statistical mechanics. The number $c_n$ of $n$-step self-avoiding walks started at a fixed vertex grows exponentially, and the base of this growth, $\mu=\lim c_n^{1/n}$, is called the connective constant of the lattice. For the hexagonal (honeycomb) lattice, Duminil-Copin and Smirnov proved that $\mu=\sqrt{2+\sqrt 2}$, confirming a 1982 prediction of Nienhuis that was obtained by Coulomb-gas methods of theoretical physics for the two-dimensional $O(n)$ model, of which the self-avoiding walk is the $n\to 0$ member. Their proof introduces a parafermionic observable: a complex-valued weighted sum over self-avoiding curves in which each curve $\gamma$ carries a weight $x_c^{\ell(\gamma)}$ times a phase $e^{-i\sigma W(\gamma)}$ depending on the total winding $W(\gamma)$ of the curve, with $\sigma=5/8$ and $x_c=1/\sqrt{2+\sqrt 2}$. This observable satisfies one half of the discrete Cauchy–Riemann relations (a discrete analogue of holomorphicity: its contour integrals around vertices vanish, yielding only about two thirds as many linear relations as there are unknowns); the other half fails at the discrete level and is only expected to emerge in the scaling limit.

The Schramm–Loewner evolution $\mathrm{SLE}_\kappa$ is the one-parameter family of random fractal curves obtained by driving the Loewner equation of complex analysis with a Brownian motion of speed $\kappa$; these are the universal scaling limits of critical two-dimensional lattice models, and chordal $\mathrm{SLE}_\kappa$ is the version joining two boundary points of a domain. Lawler, Schramm and Werner conjectured that the critical two-dimensional self-avoiding walk converges to chordal $\mathrm{SLE}_{8/3}$ and proved that a conformally invariant scaling limit of the walk would allow its critical exponents to be computed. The exponent $\gamma$ governs the subexponential correction in $c_n\sim A n^{\gamma-1}\mu^n$ (conjecturally $\gamma=43/32$, from Nienhuis), and the displacement exponent $\nu$ governs the mean square end-to-end distance $\langle|\gamma(n)|^2\rangle=n^{2\nu+o(1)}$ (conjecturally $\nu=3/4$); rigorous bounds on both remain far from these predicted values. Bridges are self-avoiding walks that are monotone in a fixed coordinate direction; their critical partition function $B_T^{x_c}$, the sum of $x_c^{\ell(\gamma)}$ over bridges of height $T$, is conjectured to decay as $T^{-1/4}$ as $T\to\infty$.

After settling the connective constant, the paper of Duminil-Copin and Smirnov formulates the remaining program as two conjectures. Conjecture 1: for any simply connected domain $\Omega\ne\mathbb{C}$ with two distinct boundary points $a,b$, the law of the random self-avoiding curve $\gamma_\delta$ joining $a_\delta$ to $b_\delta$ inside the discrete approximation $\Omega_\delta$, under the measure weighting each curve proportionally to $x_c^{\ell(\gamma)}$, converges as the mesh $\delta\to 0$ to chordal $\mathrm{SLE}_{8/3}$ in $\Omega$ from $a$ to $b$. Conjecture 2: assuming the boundary of $\Omega$ is smooth near $b$, the normalized parafermionic observable converges, $\lim_{\delta\to 0}F_\delta(z_\delta)/F_\delta(b_\delta)=(\phi'(z)/\phi'(b))^{5/8}$, where $\phi$ is a conformal map of $\Omega$ onto the upper half-plane sending $a$ to $\infty$ and $b$ to $0$ — the unique solution of a Riemann boundary value problem that the observable satisfies in discretized form. Proving Conjecture 2 would, in the authors' words, be a major step toward Conjecture 1 and the derivation of the critical exponents, since convergence of a suitable observable to a conformally covariant limit is the standard route to identifying the Loewner driving function of an ensemble of random curves. The same paper notes that the method does not extend to the loop $O(n)$ model for $0<n\le 2$ (loops obstruct the derivation of the critical point) nor to the square lattice, whose connective constant remains unknown.

Problem Statement

Prove (or refute) Conjecture 1 of Duminil-Copin–Smirnov. Let $\Omega\ne\mathbb{C}$ be a simply connected domain with two distinct boundary points $a,b$; let $\Omega_\delta\subset\delta\mathbb{H}$ be its discrete approximation on the hexagonal lattice with mesh $\delta$, with $a_\delta,b_\delta$ the vertices of $\Omega_\delta$ closest to $a,b$; and let $\gamma_\delta$ be the random self-avoiding curve from $a_\delta$ to $b_\delta$ staying in $\Omega_\delta$, under the probability measure assigning to each curve $\gamma$ a weight proportional to $x_c^{\ell(\gamma)}$ with $x_c=1/\sqrt{2+\sqrt 2}$. Show that the law of $\gamma_\delta$ converges, as $\delta\to 0$, to chordal $\mathrm{SLE}_{8/3}$ in $\Omega$ from $a$ to $b$, in the standard topology for random curves (e.g. convergence in law of parametrized curves up to reparametrization). These statements serve one overarching objective: a rigorous conformally invariant description of the critical two-dimensional self-avoiding walk, including the linked exponent predictions it is expected to deliver — $c_n\sim A n^{\gamma-1}\mu^n$ with $\gamma=43/32$, mean-square displacement $n^{2\nu}$ with $\nu=3/4$, and critical bridge decay $B_T^{x_c}\sim T^{-1/4}$. The linked exponent predictions follow from Conjecture 1 via Lawler–Schramm–Werner; conversely, a scaling limit that does not exist, or that exists but differs from chordal $\mathrm{SLE}_{8/3}$, would refute the conjecture. No finite-size, restricted-domain, or weakened-topology variant is requested: the conjecture concerns all simply connected domains as in the source.

References

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