Near-critical percolation interfaces: convergence as curves and the Loewner driving function (massive SLE$_6$)

For site percolation on the triangular lattice with the open-site probability $p$ approaching the critical value $1/2$ at the natural near-critical rate, so that the correlation length is of the order of the system size, Nolin and Werner proved that the interface has non-degenerate subsequential scaling limits, all singular with respect to the critical limit SLE$_6$ while sharing its Hausdorff dimension $7/4$. Two questions left open by that work survive to date. First, whether the exploration path actually converges as a curve to a unique scaling limit: Garban, Pete and Schramm later proved uniqueness of the near-critical configuration scaling limit in the quad-crossing topology, but explicitly did not prove curve-level convergence or the measurability of the interface as a curve. Second, the characterization of the Loewner driving function of the near-critical limit: Nolin and Werner conjectured that it couples with a Brownian motion so that $w(t)-\beta(6t)$ is an adapted increasing continuous process, and Garban, Pete and Schramm refined this into a precise drift ansatz $dA_t = c_0\lambda|d\gamma_t|^{3/4}|dW_t|$ for a 'massive SLE$_6$'. The problem is to settle both: whether the near-critical interface scaling limit converges as a curve to a unique limit, and what its driving process is — which would construct the first massive SLE in the singular regime $\kappa>4$.

Background

Site percolation on the triangular lattice $\mathbb T$ declares each site open independently with probability $p$; its critical point is $p_c = 1/2$. Under Dobrushin boundary conditions (open sites prescribed along one boundary arc, closed sites along the complementary arc), the interface or exploration path is the curve separating the open cluster attached to one arc from the closed cluster attached to the other, discovered by revealing hexagons one at a time. Smirnov's conformal invariance theorem for critical percolation, together with the work of Camia and Newman, shows that at $p=1/2$ the exploration path converges in the scaling limit to the Schramm–Loewner Evolution with parameter 6. An SLE$_\kappa$ (introduced by Schramm in 2000) is a random fractal curve in a planar domain generated by the chordal Loewner equation, the classical device that encodes a growing curve through a family of conformal maps; the curve is determined by a real-valued driving function, and SLE$_\kappa$ is the case where this driving function is $\sqrt{\kappa}$ times a standard Brownian motion. Thus the critical percolation interface is completely characterized: its driving function is Brownian motion up to the factor $\sqrt 6$.

Near-critical percolation concerns $p \to 1/2$ with the system size. Kesten's scaling relations, combined with the arm exponents computed via SLE, give the correlation length $L(p) = |p-1/2|^{-4/3+o(1)}$; the arm exponent language refers to the probability that a site is connected to distance $R$ by $k$ alternating strings of open and closed sites (the four-arm case, exponent $5/4$ for the point-to-distance probability). A central mechanism is played by pivotal sites — sites whose color change would flip a macroscopic crossing event; these are exactly the four-arm sites, and the corresponding exponent $3/4$ controls the sensitivity of the system to changes of $p$.

Nolin and Werner studied the genuinely near-critical regime, where the system size $N$ is of the order of $L(p(N))$: neither so close to $1/2$ that the limit is SLE$_6$, nor so far that the interface degenerates to the domain boundary. Using the tightness criterion of Aizenman and Burchard, they proved: subsequential scaling limits of the interface exist; every such non-degenerate limit is singular with respect to the law of SLE$_6$ (for $p>1/2$ the curve turns 'more right' than SLE$_6$, a bias detectable only by averaging over many points); the limit curves still have Hausdorff dimension $7/4$ and the same critical exponents as SLE$_6$; and the only other possible limits are SLE$_6$ itself or a degenerate boundary-hugging curve. The law of a near-critical interface limit is thus a new object: singular to, yet dimension-indistinguishable from, SLE$_6$.

Two interlocking questions were left open. (1) From subsequential limits to a limit. Camia, Fontes and Newman suggested that the near-critical scaling limit could be obtained from the critical one by opening a Poissonian family of pivotal points. Garban, Pete and Schramm realized this program at the level of configurations: they proved that near-critical percolation on $\eta\mathbb T$ with $p = p_c + \lambda r(\eta)$, where $\lambda \neq 0$ is fixed and $r(\eta) = \eta^2 (\alpha_4^\eta(\eta,1))^{-1} \asymp \eta^{3/4+o(1)}$ is the four-arm normalization ($\alpha_4^\eta$ the critical four-arm probability), converges as the mesh $\eta \to 0$ in the Schramm–Smirnov quad-crossing topology — the topology in which a percolation configuration is viewed through its crossing events for all quadrilaterals — to a unique Markovian, conformally covariant limiting configuration. However, they explicitly did not prove that the interface converges as a curve: the measurability of the exploration path as a curve in the continuum configuration is open (even at criticality, where it is Question 2.14 of their pivotal-measures paper), so the subsequential curve limits of Nolin and Werner are, in their words, 'not exactly known yet to be an actual limit'. (2) The driving function. Arm exponents suggest that a near-critical interface scaling limit $\gamma$ is almost surely a Loewner chain with some driving function $w(t)$, and the natural question is what process replaces Brownian motion. Since the near-critical curve turns 'more right' than SLE$_6$ (for $\lambda > 0$), Nolin and Werner conjectured that, after suitably extending the probability space, one can couple $w$ with a standard Brownian motion $\beta$ so that $w(t) - \beta(6t)$ is an increasing continuous process adapted to the filtration generated by $(\beta(6t))$ and $(w(t))$; they anticipated that the drift is far from regular — its derivative measure might be carried by a fractal-type set of exceptional times — so that the usual absolute-continuity intuition for Brownian motion with drift fails. Makarov and Smirnov proposed a general framework of massive SLEs as scaling limits of off-critical interfaces, with concrete candidates only for $\kappa \in \{2,3,4,16/3,8\}$, where harmonic or fermionic observables are available; this program has recently been made rigorous for loop-erased random walk ($\kappa=2$; Chelkak–Wan), near-critical dimers ($\kappa=2$; Berestycki–Haunschmid-Sibitz), the massive harmonic explorer ($\kappa=4$; Papon) and the Ising model in a vanishing magnetic field ($\kappa=3$; Papon), in all of which the massive SLE is absolutely continuous with respect to plain SLE. Percolation corresponds to $\kappa = 6 > 4$, the regime in which the massive SLE is expected to be singular with respect to SLE$_6$, so this change-of-measure technology is unavailable. Garban, Pete and Schramm refined the conjecture into a precise ansatz for the drift of a 'massive chordal SLE$_6$': $dW_t = \sqrt 6\, dB_t + dA_t$, where $W_t$ is the Loewner driving function and $B_t$ a standard Brownian motion, with $dA_t = c_0 \lambda |d\gamma_t|^{3/4}|dW_t| = c_0' \lambda |d\gamma_t|^{3/4}|dt|^{1/2}$, where $|d\gamma_t|$ is the infinitesimal Euclidean length increment of the curve — naturally formalized through the Minkowski-content measure (the fractal 'length' measure of the curve; for the critical interface this is its $7/4$-dimensional Minkowski content, cf. Holden–Li–Sun). They stressed that even constructing a Loewner chain with such a degenerate drift is a challenging open problem.

Problem Statement

Let $\mathbb T$ be the triangular lattice, and let $p(\eta) = p_c + \lambda\, r(\eta)$ with $\lambda \neq 0$ fixed and the near-critical normalization $r(\eta) = \eta^2 (\alpha_4^\eta(\eta,1))^{-1}$ (equivalently, in the $N$-box formulation of Nolin–Werner: $p(N)$ chosen so that the system size is of the order of the correlation length, $N \asymp L(p(N))$). Consider the exploration path $\gamma^\eta(\lambda)$ of near-critical site percolation on $\eta\mathbb T$ in a simply connected domain with Dobrushin boundary conditions. Determine whether $\gamma^\eta(\lambda)$ converges as $\eta \to 0$, in the standard topology on unparametrized curves used for the critical interface (uniform convergence of curve portions modulo reparametrization), to a unique scaling limit — equivalently, whether the subsequential limits of Nolin–Werner are actual limits — and characterize the Loewner driving function $w(t)$ of the limiting curve. In particular, settle:

(i) the Nolin–Werner coupling conjecture: there is an extension of the probability space carrying $w$ and a standard Brownian motion $\beta$ such that $w(t) - \beta(6t)$ is an increasing continuous process adapted to the filtration generated by $(\beta(6t))$ and $(w(t))$ when $\lambda > 0$ (decreasing when $\lambda < 0$), so that $w$ is a submartingale-driven deformation of SLE$_6$;

(ii) the Garban–Pete–Schramm drift ansatz: the drift $A_t = w(t) - \sqrt 6\, B_t$, where $B$ is a standard Brownian motion (so that $\sqrt 6\, B_t$ plays the role of $\beta(6t)$), has the form

$$dA_t = c_0 \lambda\, |d\gamma_t|^{3/4}\, |dw_t| = c_0' \lambda\, |d\gamma_t|^{3/4}\, |dt|^{1/2},$$

with $c_0, c_0'$ positive constants, $|d\gamma_t|$ the Minkowski-content length measure of the curve, and $|dw_t|$ the variation measure of the driving function itself (written $|dW_t|$ in Garban–Pete–Schramm's notation) — or determine the correct law of the driving process, including the regularity of $A_t$ and the (expected fractal) support of its derivative measure.

The task has two aims that share one scientific objective — the complete probabilistic description of the near-critical percolation interface law: (A) curve-level convergence of the interface to a unique limit, and (B) the characterization of its Loewner driving function, the latter comprising the sub-claims (i) and (ii) above. Unique curve-level convergence is the prerequisite that makes aim (B) well posed. Finer observables of the limit (complete multi-point correlation functions, multifractal spectra, the precise law of the drift beyond the stated relation) are natural consequences and further directions explicitly noted in the source work, outside the scope of the two aims above.

References

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