Open problems in the continuum bridge decomposition of restriction measures (Alberts-Duminil-Copin, 2010)

Alberts and Duminil-Copin (J. Stat. Phys. 140, 467-493, 2010) proved that chordal restriction hulls in the upper half-plane with restriction index $\alpha < 1$ - including SLE(8/3), the conjectured scaling limit of the half-plane self-avoiding walk - admit a continuum bridge decomposition: the bridge-point and bridge-height sets have Hausdorff dimension $\max(2-2\alpha,0)$, bridge times are renewal times, and the local time on bridge heights is the inverse of a stable subordinator of index $2-2\alpha$, so that the hull decomposes into a Poisson point process of irreducible continuum bridges with a sigma-finite intensity $\nu_\alpha$. The paper's final section poses seven open questions toward turning this existence theorem into a usable theory: finer properties of $\nu_\alpha$; a constructive (Loewner-driving-term, loop-soup-coupled) description of irreducible bridges; a natural length on irreducible bridges scaling from discrete walk length; a continuous analogue of Kesten's relation (a finiteness dichotomy for $\int_0^\infty \beta^{-l}\nu_\alpha(L(K)\in dl)$ with a universal critical value); a time parameterization that refreshes itself at bridge points; the use of bridge-decomposition elements to deduce half-plane self-avoiding-walk critical exponents such as the bridge-count exponent (Madras's conjecture $\beta = 7/16$); and the existence and Hausdorff dimension of bridge points for SLE($\kappa$) with $\kappa \neq 8/3$. The source poses these seven questions as open, and no resolution of any of them is known.

Background

An $N$-step self-avoiding walk (SAW) on the square lattice is a nearest-neighbour lattice path visiting no site twice; if $c_N$ counts $N$-step SAWs from the origin, submultiplicativity yields a connective constant $\mu$ with $c_N \approx \mu^N$ ($\mu \approx 2.638$ on $\mathbb{Z}+i\mathbb{Z}$). A half-plane SAW stays in $\{z : \Im z > 0\}$ after starting at the origin. A bridge is a SAW $\omega$ with $\Im\omega(0) < \Im\omega(j) \leq \Im\omega(N)$ for all $1 \leq j \leq N$, so the endpoint is the walk's (weak) maximum height. Bridges concatenate, and a bridge that is not itself a concatenation of two shorter bridges is irreducible. Kesten showed that every half-plane SAW decomposes uniquely into irreducible bridges and proved Kesten's relation $\sum_{\omega}\mu^{-|\omega|} = 1$ (the sum over irreducible bridges $\omega$), so that $\mu^{-|\omega|}$ is a probability measure on irreducible bridges; concatenating independent samples yields the infinite half-plane SAW and gives it a renewal structure: at the end of each irreducible bridge, the shifted future path is again an independent copy of the walk (see Kesten 1963 and the standard account in the book of Madras and Slade). The conjectured scaling limit of the half-plane SAW is chordal SLE(8/3) - chordal meaning that the curve grows from one boundary point to another, here from $0$ to $\infty$. The Schramm-Loewner evolution SLE($\kappa$) is the random curve obtained by driving the Loewner equation - the conformal-map evolution encoding a growing hull in the upper half-plane $\mathbb{H}$ - with standard Brownian motion of variance parameter $\kappa$. Lawler, Schramm and Werner proved that if the scaling limit of half-plane SAWs exists and is conformally invariant then it must be SLE(8/3), but existence of the limit remains unproven. A common generalization of SLE(8/3) is the family of chordal restriction measures: for $\alpha \geq 5/8$, a probability measure $\mathbb{P}_\alpha$ on unbounded hulls (closed connected sets connecting $0$ and $\infty$ in $\overline{\mathbb{H}}$, meeting $\mathbb{R}$ only at $0$, whose complement has two components) satisfying the restriction property $\mathbb{P}_\alpha(K \cap A = \emptyset) = \phi_A'(0)^\alpha$ for every hull $A$ away from the origin, where $\phi_A$ maps $\mathbb{H}\setminus A$ conformally back to $\mathbb{H}$ with $\phi_A(z) - z \to 0$ at infinity. For $\alpha = 5/8$ the restriction measure is SLE(8/3) itself; for $\alpha = 1$ it is the filled-in Brownian excursion; in general it is obtained by attaching to an SLE($\kappa$) curve (with $\kappa = 6/(2\alpha+1)$) all the loops it intersects from an independent Brownian loop soup of intensity $\lambda = (8-3\kappa)\alpha$ - the Brownian loop soup being the Poissonian conformally invariant ensemble of unrooted Brownian loops of Lawler and Werner. Alberts and Duminil-Copin proved that these restriction hulls carry a continuum bridge decomposition when $\alpha < 1$. A height $L > 0$ is a bridge height of an unbounded hull $K$ if the horizontal line $\{y = L\}$ intersects $K$ in exactly one point $z$ (a bridge point); $C$ and $D$ denote the sets of bridge points and bridge heights. They showed that $\dim_{\mathrm{H}} C = \dim_{\mathrm{H}} D = \max(2-2\alpha, 0)$ almost surely (Hausdorff dimension being the standard fractal dimension; the sets are empty exactly when $\alpha \geq 1$), that bridge times are renewal times for the growing hull, and that there is a local time on bridge heights (an additive functional accumulating only on that set) whose inverse is a stable subordinator of index $2-2\alpha$ - a subordinator being an increasing pure-jump Lévy process, here recording the heights swept between successive bridge points - so that the hull decomposes into a Poisson point process of irreducible continuum bridges with a sigma-finite intensity measure $\nu_\alpha$. The only quantitative property of $\nu_\alpha$ identified is the height tail $\nu_\alpha(\text{height} > L) = c_\alpha L^{2\alpha-2}$. The construction is the exact analogue of Itô's excursion decomposition of Brownian motion at its zeros, of Virág's Brownian beads (an analogous decomposition of planar Brownian hulls), and of Dubédat's excursion decompositions of SLE-type hulls at cutpoints - points whose removal disconnects the hull; every bridge point is a cutpoint but not conversely. Two neighbouring bodies of results frame the open questions. First, the geometry of special point sets of SLE curves: Beffara proved the SLE($\kappa$) curve has Hausdorff dimension $1+\kappa/8$, and Alberts and Sheffield computed the dimension of the set where the SLE($\kappa$) curve meets the real line, and Miller and Wu computed the dimensions of the double and cut points of chordal SLE($\kappa$) for $\kappa > 4$ (cut points of a set being points whose removal disconnects it) - but bridge points have only been analysed at $\kappa = 8/3$, inside restriction hulls. Second, the natural parametrization of Lawler and Sheffield: for $\kappa < 8$ (under a technical condition they verify for $\kappa < 4(7-\sqrt{33}) \approx 5.02$) there is a canonical measure on the SLE$_\kappa$ curve that transforms like $d$-dimensional volume under conformal maps, with $d = 1+\kappa/8$, giving the curve a natural length in the same sense in which discrete curve length is expected to scale; Lawler and Zhou proved the natural parametrization is well defined for all $\kappa < 8$, Lawler and Rezaei proved that it is in fact given by the Minkowski content of the curve (the limiting renormalized volume of its $\epsilon$-neighbourhoods), and Alberts, Kozdron and Masson studied the convergence of loop-erased random walk to SLE(2) in this parametrization. Whether any such length restricts to a natural length on irreducible bridges, or refreshes at bridge times, is open.

Problem Statement

The problem is to complete the continuum bridge-decomposition theory of restriction measures, bringing it from an existence theorem to a theory with the analytic power of its discrete prototype, Kesten's bridge decomposition of half-plane self-avoiding walks. Alberts and Duminil-Copin explicitly posed the following interlinked questions, and the problem is to resolve them as stated, within the framework of the source (restriction hulls $\mathbb{P}_\alpha$ with $\alpha \in [5/8, 1)$, SLE($\kappa$) curves, bridge points/heights/times as defined there, and the irreducible-bridge intensity $\nu_\alpha$):

1. Structure of $\nu_\alpha$: what properties of the irreducible-bridge intensity $\nu_\alpha$ beyond sigma-finiteness and the height tail $\nu_\alpha(\text{height} > L) = c_\alpha L^{2\alpha-2}$ can be derived - finer distributional or structural properties of continuum irreducible bridges?
2. Constructive description: is there a constructive way of building irreducible bridges - for SLE(8/3), a Loewner-equation driving term that outputs irreducible bridges (perhaps with at least some specified vertical height), and for general restriction measures with $\alpha < 1$, a driving term for the Loewner equation combined with the Brownian loop soup producing irreducible bridges of restriction hulls?
3. Natural length: is there a natural "length" that can be put on irreducible bridges, expected to be the scaling limit of the length of discrete self-avoiding walks, presumably related to the Lawler-Sheffield natural time parameterization of SLE(8/3)?
4. Continuous analogue of Kesten's relation: if $L(K)$ denotes the natural length of an irreducible bridge, is the integral

$$\int_0^\infty \beta^{-l}\, \nu_\alpha(L(K) \in dl)$$

finite for $\beta < \mu$ but infinite for $\beta > \mu$, for some universal $\mu$, and what is the behaviour at this critical $\mu$? 5. Self-refreshing parameterization: can the restriction hulls be time-parameterized so that the time parameterization also refreshes itself at bridge points - i.e. the time-parameterized future hull, shifted at a bridge time, is independent of the past and has the same parameterized law - as the step-counting parameterization does for half-plane SAWs? 6. Critical exponents: can some element of the bridge decomposition be used to prove the existence of, or at least heuristically deduce, critical exponents for half-plane SAWs or SAW bridges - for instance the conjecture that the number of $N$-step SAW bridges grows asymptotically like $N^{-\beta}u^N$ as $N \to \infty$, for the same growth constant $u = \mu$ as in $c_N \approx \mu^N$ and some unknown constant $\beta$, where Neal Madras has conjectured $\beta = 7/16$? 7. Bridge points for other SLE($\kappa$): do bridge heights and lines exist for SLE($\kappa$) curves for values of $\kappa$ different from $8/3$, and if so, what is the Hausdorff dimension of the bridge-point set $C$ and the bridge-height set $D$, and how does it depend on $\kappa$? The known anchors are: dimension $1$ at $\kappa = 0$ (the curve is a vertical line), $3/4$ at $\kappa = 8/3$, and necessarily $0$ at $\kappa = 4$ (the SLE(4) curve comes arbitrarily close to the real line). The source conjectures that the dimensions of $C$ and $D$ are always equal and are a continuous function of $\kappa$ that is strictly decreasing while it is positive; the smallest $\kappa$ for which the dimension is zero is unknown (somewhere between $8/3$ and $4$); a lower bound $\dim_{\mathrm{H}} C \geq 3 - 6/\kappa$ holds for $2 \leq \kappa \leq 8/3$. The seven questions are the full scope of this objective: completing the continuum bridge-decomposition theory means resolving them. Each question is posed separately and independently, so resolving one leaves the others open; the seven nevertheless share one objective - making the continuum decomposition as usable as the discrete one - and are interdependent: a natural length (item 3) is the prerequisite for the Kesten-type dichotomy (item 4) and for transferring continuum quantities to discrete exponents (item 6), while finer structure of $\nu_\alpha$ (item 1) and a constructive description (item 2) are expected to be linked.

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