Loop Soup Research Articles

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

It has been recently understood [9, 24, 30] that for a general class of percolation models on $\mathbb{Z} ^{d}$ satisfying suitable decoupling inequalities, which includes i.a. Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT. In this paper we consider the random walk loop soup on $\mathbb{Z} ^{d}$ in dimensions $d\geq 3$. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since the required decoupling inequalities are not valid. We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang [6] by a model specific approximation method, which does not apply to the vacant set.) Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups

We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall.

Universal statistics of vortex tangles in three-dimensional random waves

The tangled nodal lines (wave vortices) in random, three-dimensional wavefields are studied as an exemplar of a fractal loop soup. Their statistics are a three-dimensional counterpart to the characteristic random behaviour of nodal domains in quantum chaos, but in three dimensions the filaments can wind around one another to give distinctly different large scale behaviours. By tracing numerically the structure of the vortices, their conformations are shown to follow recent analytical predictions for random vortex tangles with periodic boundaries, where the local disorder of the model ‘averages out’ to produce large scale power law scaling relations whose universality classes do not depend on the local physics. These results explain previous numerical measurements in terms of an explicit effect of the periodic boundaries, where the statistics of the vortices are strongly affected by the large scale connectedness of the system even at arbitrarily high energies. The statistics are investigated primarily for static (monochromatic) wavefields, but the analytical results are further shown to directly describe the reconnection statistics of vortices evolving in certain dynamic systems, or occurring during random perturbations of the static configuration.

Spin systems from loop soups

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.

Intersection Local Times, Loop Soups and Permanental Wick Powers

Several stochastic processes related to transient Levy processes with potential densities u(x,y) = u(y x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures V endowed with a metric d. Su- cient conditions are obtained for the continuity of these processes on (V,d). The processes include n-fold self-intersection local times of tran- sient Levy processes and permanental chaoses, which are 'loop soup n- fold self-intersection local times' constructed from the loop soup of the Levy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional proper- ties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

Simple CLE in doubly connected domains

Nous étudions l’ensemble des boucles conformes ($\operatorname{CLE}_{\kappa}$) dans des domaines connexes du type : anneau, disque percé et plan percé. Nous considérons les cas $\operatorname{CLE}_{\kappa}$ pour lesquels les boucles sont simples, i.e. $\kappa\in(8/3,4]$. Dans (Ann. of Math. (2) 176 (2012) 1827–1917), l’ensemble $\operatorname{CLE}$ dans le disque unité est introduit et construit comme la collection de frontières extérieures des amas les plus excentrés de la soupe de boucles Browniennes. Dans le cas du disque unité, n’importe quel point intérieur est presque sûrement entouré par une boucle du $\operatorname{CLE}$. L’ensemble des points qui ne sont entourés par aucune boucle a une mesure de Lebesgue nulle presque sûrement. Dans notre article, le $\operatorname{CLE}$ dans un anneau est construit de façon similaire : il s’agit de la collection de frontières extérieures des amas de la soupe de boucles Browniennes conditionnés sur l’évènement qu’il n’existe pas d’amas séparant les deux composantes de la frontière de l’anneau. Dans le cas du disque percé, le $\operatorname{CLE}$ correspond au conditionnement par le fait que l’origine n’est pas entourée par une boucle. Dans le cas du plan percé, le conditionnement est tel que l’origine et l’infini ne sont pas entourés. Nous construisons et étudions ces trois types de $\operatorname{CLE}$ et les processus d’exploration correspondants.

Supercritical loop percolation on [formula omitted] for [formula omitted

We consider supercritical percolation on Zd (d≥3) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all d≥3 and all supercritical intensities despite polynomial decay of correlations.

From loop clusters and random interlacements to the free field

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops ("loop soup") of parameter one-half associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field on this network. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter one-half do not percolate on periodic lattices. We also construct a coupling between the random interlacement on $\mathbb{Z}^{d}$, $d\geq 3$, introduced by Sznitman, and the Gaussian free field on the lattice such that the set of vertices visited by the interlacement is contained in a level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement. Both in the case of loops and of the random interlacement, the couplings are constructed by replacing discrete graphs by metric graphs. Le Jan's and Sznitman's isomorphism theorems between the Gaussian free field and the occupation field of trajectories can be extended to the metric graph setting on which the intermediate value principle for continuous fields holds.

Loop percolation on discrete half-plane

We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of percolation at intensity 1/2 was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.

Conformal correlation functions in the Brownian loop soup

We define and study a set of operators that compute statistical properties of the Brownian loop soup, a conformally invariant gas of random Brownian loops (Brownian paths constrained to begin and end at the same point) in two dimensions. We prove that the correlation functions of these operators have many of the properties of conformal primaries in a conformal field theory, and compute their conformal dimension. The dimensions are real and positive, but have the novel feature that they vary continuously as a periodic function of a real parameter. We comment on the relation of the Brownian loop soup to the free field, and use this relation to establish that the central charge of the loop soup is twice its intensity.

Random walk loop soups and conformal loop ensembles

AbstractThe random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm–Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, “large” lattice loops from the random walk loop soup converge to “large” loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of “large” lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of “large” lattice loops converges to CLE.

Permanental fields, loop soups and continuous additive functionals

A permanental field, $\psi=\{\psi(\nu),\nu\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $\psi$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates $\psi$ to continuous additive functionals of $X$ (continuous in $t$), $L=\{L_t^{\nu},(\nu ,t)\in {\mathcal{V}}\times R_+\}$. Sufficient conditions are obtained for the continuity of $L$ on ${\mathcal{V}}\times R_+$. The metric on ${\mathcal{V}}$ is given by a proper norm.

Markovian loop soups: permanental processes and isomorphism theorems

We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.

Universal Behavior of Connectivity Properties in Fractal Percolation Models

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension $d\geq2$. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for $d=2$) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter $\lambda$. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of $\lambda$ that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions $d\geq2$. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for $d=2$ we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.

Stochastic Analysis

Stochastic Analysis

Random walk loop soup

The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlos, Major, and Tusnady. To make the paper self-contained, we include a proof of the approximation result that we need.

SLEs as boundaries of clusters of Brownian loops

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLE κ -type curves where κ∈(8/3,4] and c related by the usual relation c=(3 κ−8)(6− κ)/2 κ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLE κ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory. To cite this article: W. Werner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).