Random soup carpets beyond symmetry: the converse to subcriticality, critical behaviour at the carpet phase transition, and general shape laws

Nacu and Werner (2011) studied carpets obtained by removing the interiors of a Poissonian, translation- and scale-invariant 'soup' of planar curves, and proved under thinness and subcriticality assumptions that the carpet's Hausdorff dimension is deterministic and equal to its expectation dimension, with small-intensity expansion $d(c)=2-c\beta(\pi)+o(c)$ ; for the Brownian loop soup this completed the exact dimension theory of conformal loop ensemble carpets. Their paper explicitly leaves open the removal of the auxiliary assumptions of this theory. First, the converse to subcriticality: does thinness plus non-triviality of the carpet imply that all clusters of the soup are bounded, without rotation-invariance or other symmetry of the shape measure? Second, the critical behaviour at the carpet's critical intensity $c_0$ for general shape measures: is the approximate carpet non-empty exactly at $c=c_0$ without symmetry assumptions? Later work resolves adjacent questions — Broman and Camia prove connectivity at the critical point for general continuum scale-invariant fractal models, Broman determines the exact existence-phase-transition threshold for general measures, and Teixeira and Ungaretti rigorously develop the scale-invariant stick soup and prove the converse direction there while stating that the coincidence of the carpet-nontriviality and cluster-percolation thresholds remains 'not clear' — but the two general questions above survive as an open core.

Background

Fractal (Mandelbrot) percolation is a classical random Cantor-set construction: subdivide the unit square into dyadic squares, delete each independently with probability $p$ , and iterate across scales. The surviving set is governed by a Galton-Watson tree (a branching-process model of generation survival), and Chayes–Chayes–Durrett (1988) analysed its connectivity properties at and near the critical parameter. Nacu and Werner (2011) studied a fully Poissonian, exactly scale- and translation-invariant analogue. The intensity measure $\mu$ is a locally finite measure on compact planar curves, invariant under dilations and translations and not concentrated on parallel line segments; every such $\mu$ decomposes into a location component (planar Lebesgue measure), a scale component ( $d\rho/\rho$ up to convention), and a finite shape measure $\pi$ on curves of diameter 1 with $\pi(|\gamma|^2)<\infty$ . A Poisson point process of curves with intensity $c\mu$ — a random countable collection of curves in which the numbers of curves falling in disjoint regions are independent Poisson random variables with means given by the intensity measure — is called a random soup; it removes the interiors of its curves from a bounded simply connected domain $D$ . The remaining set is $F=D\setminus\bigcup_j O_j$ (where $O_j$ is the bounded component of the complement of the $j$ -th curve); the carpet $G$ is the set of points of $D$ that can be connected to $\partial D$ through $F$ (paths may touch curves but not enter interiors); the approximate carpet $C=\bigcap_\epsilon C_\epsilon$ is a measurable approximation that ignores curves within $\epsilon$ of a point, and it differs from $G$ only by the union of the curves. Two curves of the soup are connected if they intersect and so do their interiors; the equivalence classes are the clusters of the soup. The soup is thin if only finitely many curves meet both boundary circles of any fixed annulus, equivalently if a certain integral of the mean $r$ -neighbourhood areas of the shapes converges; thinness prevents pathologically dense soups. A 0–1 argument shows that every full-plane soup is in exactly one of two regimes: either all clusters are almost surely bounded, or there is a unique dense unbounded cluster. The theory is developed under the subcriticality assumptions: SA1, all full-plane clusters are almost surely bounded, and the slightly stronger SA2, with positive probability some closed loop surrounds the origin crossing no loop of the soup. Subcriticality implies thinness and non-empty carpets, and it is the converse of this implication that is at issue. Under thinness and subcriticality, Nacu–Werner proved by second-moment estimates (comparing the mean and second moment of the number of small balls needed to cover the set) that the carpet's Hausdorff dimension — the standard notion of fractal dimension, defined via coverings by sets of arbitrarily small diameter — is non-random and equal to its expectation dimension (the exponent governing the mean number of $\epsilon$ -balls needed to cover it), and that as the intensity $c\to 0^+$ the dimension satisfies $d(c)=2-c\beta(\pi)+o(c)$ , where $\beta(\pi)=\pi(A(\gamma))$ is the expected area surrounded by a shape. Varying $c$ , the approximate carpet is almost surely empty for large $c$ and almost surely non-empty for small $c$ , so there is a critical intensity $c_0$ separating the two regimes; Nacu–Werner note that non-emptiness at $c=c_0$ itself should follow by adapting fractal-percolation arguments 'at least when $\pi$ is invariant under some rotations'. The central special case is the Brownian loop soup of Lawler and Werner, a Poissonian ensemble of planar Brownian loops with a conformally invariant intensity measure; through the work of Sheffield and Werner connecting loop-soup cluster boundaries to conformal loop ensembles (CLE, the canonical conformally invariant collections of nested loops whose boundaries are Schramm–Loewner evolution curves), the carpet is non-trivial exactly for $0<c\le 1$ , its expectation dimension was computed by Schramm–Sheffield–Wilson, and Nacu–Werner's theorem made this the exact almost-sure dimension; Miller–Sun–Wilson completed the analogous formula for the CLE gasket family (the set of points surrounded by no loop of a CLE). For general shape measures no conformal invariance is available. Two related phase transitions of the same model class are understood: Broman–Camia (2010) proved that for general continuum scale-invariant fractal percolation models the vacant set has connected components larger than a point at its connectivity threshold $\lambda_c$ , without symmetry assumptions (in the plane this yields a unique unbounded component); and Broman (2020) determined the exact existence threshold $\lambda_e=d\log 2/\mu(A_1)$ above which the remaining set is empty for all measures satisfying weak assumptions, proving emptiness at $\lambda_e$ for a large subclass including the Brownian loop soup (building on Broman–Jonasson–Tykesson for ball models). These thresholds concern the full-plane vacant set; the carpet threshold $c_0$ and the subcriticality regime of the soup itself are the subject of the present problem.

Problem Statement

Work in the framework of Nacu–Werner (2011). Let $\mu$ be a locally finite, translation-invariant and scale-invariant measure on compact planar curves that is not one-dimensional, with finite shape measure $\pi$ ; for intensity $c>0$ let the soup $\Gamma(c)$ be the Poisson point process of curves with intensity $c\mu$ . Thinness (only finitely many curves meet both boundary circles of any fixed annulus), the carpet $G$ and the approximate carpet $C$ in a bounded simply connected domain, the clusters of the soup, the subcriticality assumptions SA1 (all full-plane clusters are almost surely bounded) and SA2 (with positive probability a closed loop surrounds the origin crossing no loop of the soup), and the critical intensity $c_0$ separating the regimes where the approximate carpet is almost surely empty and almost surely non-empty are all as defined in that paper. The common objective is to remove the auxiliary symmetry and criticality assumptions under which the Nacu–Werner carpet theory is rigorously established, i.e. to give the complete phase description of general scale-invariant soups. The problem has two linked components. (1) Converse to subcriticality: prove, without any rotation-invariance or other symmetry assumption on $\mu$ , that thinness of the soup together with non-triviality of the carpet (non-emptiness with positive probability in some bounded domain) implies subcriticality (SA1 and SA2) at the same intensity — or construct an explicit counterexample, a thin soup with non-trivial carpet whose clusters are nevertheless unbounded (equivalently, form a unique dense cluster). In threshold form: determine whether the carpet-nontriviality threshold $c_0$ coincides with the cluster-percolation threshold, i.e. whether there can be an intermediate regime of intensities with unbounded (percolating) clusters but a non-trivial carpet; this coincidence is known for the Brownian loop soup and was proved for the stick soup only in the one-sided form (non-percolation below the box-crossing threshold) by Teixeira–Ungaretti, who state that the full coincidence remains unclear there. (2) Critical behaviour at $c_0$ : for arbitrary shape measures $\pi$ without symmetry assumptions, determine whether the approximate carpet is non-empty or empty exactly at $c=c_0$ . (In the Brownian loop soup case the critical intensity is $c=1$ , the carpet is non-trivial there, and exact dimensions and crossing exponents are known via conformal invariance; nothing of this sort is known for general $\pi$ .)

References

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