Convergence of the two-dimensional self-avoiding walk to chordal $\mathrm{SLE}_{8/3}$ and the conjectured intersection exponents

Lawler and Werner (2000) constructed a class of conformally invariant measures on planar 'oriented bridges' — curves crossing the annuli of a disk, invariant under conformal maps and under restriction of the domain — and proved a universality theorem: every measure in this class has the same intersection exponents as a pack of $\alpha$ planar Brownian motions, where $\alpha\ge 1$ is its crossing exponent. They asked whether the class contains non-Brownian measures, whether the scaling limits of the two-dimensional self-avoiding walk (SAW) and of critical percolation clusters belong to the class or its universality class, and how to prove the exact conjectured exponent values. The exact exponent values were settled in 2001–2002 by Lawler, Schramm and Werner using Schramm–Loewner evolution (SLE); the classification question was settled by the conformal restriction program (chordal case 2003; radial, trichordal and generalized cases 2015–2021; uniqueness for central charge $c\le 1$ in 2024–2026), which also dissolved the 2000 paper's '5/8 of a Brownian motion' paradox by showing that chordal $\mathrm{SLE}_{8/3}$ is precisely the restriction measure with exponent $5/8$; the percolation half of the scaling-limit question is rigorous for site percolation on the triangular lattice. The remaining open problem is the self-avoiding-walk half: prove that the SAW on a fixed planar lattice has a conformally invariant scaling limit identified with chordal $\mathrm{SLE}_{8/3}$ (equivalently, the restriction measure $P_{5/8}$), or at least that the SAW intersection and disconnection exponents take the predicted Brownian-universality values (the quadratic formulas conjectured by Duplantier and Kwon). This remains open for every fixed lattice; the analogous convergence is proven only when the underlying graph is itself random.

Background

A self-avoiding walk (SAW) of $N$ steps on a planar lattice such as $\mathbb{Z}^2$ or the hexagonal lattice is a nearest-neighbour lattice path that visits no vertex twice; it is the standard lattice model of a linear polymer. The number of $N$-step walks grows exponentially at a lattice-dependent rate governed by the connective constant (the exponential growth rate $\mu$ of the number of walks, so that the count grows roughly like $\mu^N$), and a central prediction of two-dimensional critical physics is that, in the scaling limit ($N\to\infty$ with the lattice mesh tending to $0$), the walk becomes a conformally invariant random curve whose critical exponents — the power-law decay rates of quantities such as the probability that two independent long walks never intersect — are universal, i.e. independent of the lattice and equal to values predicted by conformal field theory, the continuum theory describing critical scaling limits whose universality classes are labelled by a parameter $c$ called the central charge.

For planar Brownian motion the analogous exponents are rigorously defined: if $n$ and $p$ independent planar Brownian motions are started from two distinct points, the probability that the two packets of paths avoid each other up to time $t$ decays like $t^{-\xi(n,p)/2}$, which defines the Brownian intersection exponents $\xi(n,p)$; Lawler and Werner (1999) showed that these exponents extend to 'packs' containing a non-integer number of Brownian motions. Lawler and Werner (J. Eur. Math. Soc. 2 (2000) 291–328) introduced a class of infinite measures on 'oriented bridges' — curves crossing the concentric annuli of a disk — characterized by conformal invariance and by invariance under restriction of the domain; these are the CCI measures. Each such measure has a crossing exponent $\alpha$: the mass of bridges crossing an annulus of modulus $L$ decays like $e^{-\alpha L}$. Their universality theorem states that the intersection exponents of any CCI measure are exactly the Brownian intersection exponents of a pack of $\alpha$ Brownian motions, so that proving the exponent values for one member of the class would give them for all. Their Proposition 1 shows $\alpha\ge 1$ for every CCI measure, and the Brownian excursion measure (Brownian motion conditioned to stay in a half-plane), together with unions of non-intersecting excursions, realizes every $\alpha\in[1,\infty)$. Their introduction poses three open questions: (i) are there non-Brownian measures in the class; (ii) do the scaling limits of the two-dimensional SAW and of critical percolation clusters converge to measures in the class, or at least land in the same universality class so that their exponents are Brownian exponents; (iii) how can one prove the exact conjectured values of the Brownian exponents (the quadratic formulas predicted by Duplantier and Kwon)? Section 5 develops the heuristics that a single SAW should count as '5/8 of a Brownian motion' for exponent bookkeeping — although no CCI measure can have $\alpha=5/8$ — that a critical percolation cluster should count as a union of two Brownian paths, and that a 'locality' property (discrete dynamics insensitive to the distant boundary) combined with conformal invariance should correspond to central charge $0$ and force the Brownian outer-boundary multifractal spectrum (the family of local dimensions of the harmonic measure, i.e. the landing distribution of independent Brownian paths, on the outer boundary).

The subsequent development of Schramm–Loewner evolution reshaped all three questions. An $\mathrm{SLE}_\kappa$ is a random non-self-crossing planar growth process, defined through Loewner's equation — the classical representation of a growing planar curve by a one-dimensional driving function, here a Brownian motion of variance parameter $\kappa$ — and characterized (under mild regularity hypotheses) by conformal invariance together with a domain-Markov property (restarted from any intermediate time in the remaining domain, the curve has again the same law); Schramm (1999) introduced it precisely to capture the conjectured scaling limits of critical lattice models. Lawler, Schramm and Werner used $\mathrm{SLE}_6$ to compute the exact values of all Brownian intersection exponents, settling question (iii) in 2001–2002, including the disconnection exponents and the conjecture that the Brownian frontier (the outer boundary of a planar Brownian path) has Hausdorff dimension $4/3$. The conformal restriction program then settled question (i) and dissolved the $5/8$ paradox: Lawler–Schramm–Werner (2003) classified all random closed subsets of the upper half-plane joining $0$ to $\infty$ whose law is invariant under restriction of the domain — the chordal restriction measures $P_\alpha$, characterized by $P_\alpha[K\subset D]=\Phi_D'(0)^\alpha$ for conformal maps $\Phi_D$ of the half-plane onto subdomains fixing $0$ and $\infty$. These measures exist if and only if $\alpha\ge 5/8$; $P_{5/8}$ is exactly the trace of chordal $\mathrm{SLE}_{8/3}$ and is the unique member supported on simple curves; $P_\alpha$ for $\alpha>5/8$ is an $\mathrm{SLE}_\kappa$ curve decorated by an independent Poisson cloud of Brownian 'bubbles' (small excursion loops attached to the boundary), with $\alpha(\kappa)=(6-\kappa)/(2\kappa)$; the filled-in Brownian excursion is $P_1$; and all these random sets have boundary dimension $4/3$. The class therefore contains an explicit one-parameter family of genuinely non-Brownian measures, and the heuristic value $5/8$ attached to the SAW is exactly the restriction exponent of $\mathrm{SLE}_{8/3}$: the SAW scaling limit is expected to satisfy the chordal restriction property rather than the strictly stronger annular CCI property of the 2000 paper (whose crossing exponents are forced to be at least $1$) — a loophole the 2000 paper itself anticipated when it noted that strict CCI invariance might fail in scaling limits through lattice-dependent multiplicative factors. The classification was extended to the radial case (sets joining an interior point to the boundary; Wu 2015), the trichordal case (three boundary points; Qian 2018), and generalized two-parameter radial families built from 'hypergeometric SLEs' (Qian 2021), and uniqueness — formulated as the Kontsevich–Suhov conjectures — was proved for central charge $c\le 1$ by Baverez–Jego (2024, via a rigorous conformal field theory of SLE loop measures) and by Cai–Gao (2026).

For percolation, question (ii) is a theorem on the triangular lattice: Smirnov (2001) proved conformal invariance and Cardy's formula (the predicted exact crossing probabilities between boundary arcs of a domain) for critical site percolation, Camia–Newman (2007) proved convergence of the exploration path (the interface between open and closed sites, revealed by probing the lattice along a boundary) to $\mathrm{SLE}_6$, and the $\mathrm{SLE}_6$-based computations of Lawler–Schramm–Werner give the predicted exponents, so that Lawler and Werner's identification of one percolation cluster with a union of two Brownian paths holds rigorously at the exponent level for that model. For the self-avoiding walk on fixed lattices nothing comparable is known: the main rigorous results are the exact connective constant of the hexagonal lattice (Duminil-Copin–Smirnov 2012) and soft bounds, and Gwynne–Miller (2021) proved convergence to $\mathrm{SLE}_{8/3}$ only in the setting where the lattice itself is random (a uniform infinite quadrangulation — a random planar graph all of whose faces are quadrilaterals — decorated by a SAW, converging to $\mathrm{SLE}_{8/3}$ on a $\sqrt{8/3}$-Liouville quantum gravity surface, i.e. a random fractal Riemann surface coupled to the walk). The conjecture that the infinite half-plane self-avoiding walk converges to chordal $\mathrm{SLE}_{8/3}$, stated in the 2003 chordal restriction paper, remains open for every fixed planar lattice, and it is this conjecture — together with the exponent identifications it would deliver — that is the open problem addressed here.

Problem Statement

Prove that the two-dimensional self-avoiding walk on a fixed planar lattice (the hexagonal and square lattices being the standard targets) has a conformally invariant scaling limit, and identify it with chordal $\mathrm{SLE}_{8/3}$ — equivalently, with the chordal restriction measure $P_{5/8}$, the unique probability measure on closed subsets of the upper half-plane joining $0$ to $\infty$ that satisfies conformal restriction with avoidance probability $P_{5/8}[K\subset D]=\Phi_D'(0)^{5/8}$ for every simply connected subdomain $D$ of the upper half-plane, where $\Phi_D$ denotes a conformal map of the upper half-plane onto $D$ fixing $0$ and $\infty$. Concretely, for the hexagonal or square lattice: show that the uniform measure on self-avoiding excursions started at a boundary point (walks started at the boundary and conditioned to remain in the domain), renormalized according to the lattice connective constant, converges in law — in a topology strong enough to imply convergence of crossing and non-intersection probabilities — to chordal $\mathrm{SLE}_{8/3}$ from $0$ to $\infty$ in the upper half-plane. Alternatively, and this weaker form is explicitly admitted by the question of Lawler and Werner (which asks whether the discrete measures converge to the class 'or at least in the same universality class'), prove that the intersection and disconnection exponents of self-avoiding walks on a fixed lattice — respectively, the power-law decay rates of the probabilities that several independent walks avoid one another, and of the probability that a single walk's path separates a fixed distant point from infinity — exist and equal the corresponding Brownian-universality values with pack size $5/8$, i.e. the quadratic formulas predicted by Duplantier and Kwon and stated as equations (20)–(21) in Lawler–Werner (2000), so that a single self-avoiding walk rigorously counts as '5/8 of a planar Brownian motion' in the exponent bookkeeping. A solution of the convergence form would also settle the companion identifications conjectured by Lawler and Werner: that the self-avoiding-walk path has fractal dimension $4/3$ (matching the Brownian frontier) and that its multifractal spectrum (the family of local dimensions of the harmonic measure on its outer boundary) agrees with the Brownian one. The question is posed for fixed lattices: the known convergence result of Gwynne–Miller for self-avoiding walks on random quadrangulations addresses a different (random-graph) setting and does not answer it, and numerical confirmation does not constitute a solution. The percolation analogue of this question is already rigorous for site percolation on the triangular lattice; proving percolation conformal invariance on other lattices is an adjacent but separate open problem.

References

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