Random Interlacements Research Articles

Gaussian Random Permutation and the Boson Point Process

We construct an infinite volume spatial random permutation \(({{\textsf {X}}},\sigma )\), where \({{\textsf {X}}}\subset {\mathbb {R}}^d\) is locally finite and \(\sigma :{{\textsf {X}}}\rightarrow {{\textsf {X}}}\) is a permutation, associated to the formal Hamiltonian $$\begin{aligned} H({{\textsf {X}}},\sigma ) = \sum _{x\in {{\textsf {X}}}} \Vert x-\sigma (x)\Vert ^2. \end{aligned}$$The measures are parametrized by the point density \(\rho \) and the temperature \(\alpha \). Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (Phys Rev 91:1291–1301, 1953). Let \(\rho _c=\rho _c(\alpha )\) be the critical density for Bose–Einstein condensation in Feynman’s representation. Each finite cycle of \(\sigma \) induces a loop of points of \({{\textsf {X}}}\). For \(\rho \le \rho _c\) we define \(({{\textsf {X}}}, \sigma )\) as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (Probab Theory Related Fields 128(4):565–588, 2004). We also construct Gaussian random interlacements, a Poisson process of doubly infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (Ann Math 2 171(3):2039–2087, 2010). For \(d\ge 3\) and \(\rho >\rho _c\) we define \(({{\textsf {X}}},\sigma )\) as the superposition of independent realizations of the Gaussian loop soup at density \(\rho _c\) and the Gaussian random interlacements at density \(\rho -\rho _c\). In either case we call \(({{\textsf {X}}}, \sigma )\) a Gaussian random permutation at density \(\rho \) and temperature \(\alpha \). The resulting measure satisfies a Markov property and it is Gibbs for the Hamiltonian H. Its point marginal \({{\textsf {X}}}\) has the same distribution as the boson point process introduced by Shirai-Takahashi (J Funct Anal 205(2):414–463, 2003) in the subcritical case, and by Tamura-Ito (J Funct Anal 243(1): 207–231, 2007) in the supercritical case.

Random interlacements for vertex-reinforced jump processes

We introduce random interlacements for transient vertex-reinforced jump processes on a general graph G. Using increasing finite subgraphs Gn of G with wired boundary conditions, we show convergence of the vertex-reinforced jump process on Gn observed in a finite window to the random interlacement observed in the same window.

On (Non-)Monotonicity and Phase Diagram of Finitary Random Interlacement.

In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, for all , FRI is NOT stochastically monotone as fiber length increases. At the same time, numerical evidence still strongly supports the existence and uniqueness of a critical fiber length, which is estimated theoretically and numerically to be an inversely proportional function with respect to system intensity.

Some rigorous results on the phase transition of finitary random interlacements

In this paper, we show several rigorous results on the phase transition of Finitary Random Interlacements (FRI). For the high intensity regime, we show the existence of a critical fiber length, and find its exact asymptotic as intensity goes to infinity. At the same time, our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity.

On the percolative properties of the intersection of two independent interlacements

We prove the existence of non-trivial phase transitions for the intersection of two independent random interlacements and the complement of the intersection. Some asymptotic results about the phase curves are also obtained. Moreover, we show that at least one of these two sets percolates in high dimensions.

Percolation for the Finitary Random interlacements

In this paper, we prove a phase transition in the connectivity of Finitary Random interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d$, with respect to the average stopping time. For each $u>0$, with probability one $\mathcal{FI}^{u,T}$ has no infinite connected component for all sufficiently small $T>0$, and a unique infinite connected component for all sufficiently large $T<\infty$. This answers a question of Bowen in the special case of $\mathbb{Z}^d$.

Solidification of porous interfaces and disconnection

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian motion starting in the compact set. As an application of our results, we substantially strengthen the results of [22], and obtain when d \geq 3 , large deviation upper bounds on the probability that simple random walk in \mathbb Z^d , or random interlacements in \mathbb Z^d , when their vacant set is in a strongly percolative regime, disconnect the discrete blow-up of a regular compact set from the boundary of the discrete blow-up of a box containing the compact set in its interior. Importantly, we make no convexity assumption on the compact set. It is plausible, although open at the moment, that the upper bounds that we derive in this work match in principal order the lower bounds of [15] in the case of random interlacements, and of [14] for the simple random walk.

Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection

We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the occupation-time profile undergoes an entropic push governed by a specific function depending on $A$. Similar entropic repulsion phenomena conditioned on disconnection by level-sets of the discrete Gaussian free field on $\mathbb{Z}^d$, $d \geq 3$, have been obtained by the authors in arxiv:1808.09947. Our proofs rely crucially on the `solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.

Design and fabrication of lab-scale wet-laid equipment to develop wet-laid nonwoven for the application of absorbent core of disposable nursing pad

This article deals with a newly developed lab-scale equipment fabricated to produce a wet-laid nonwoven, for the application of absorbent core of the disposable nursing pad. This equipment was designed based on the principle of filtration and thickening of fiber, such that it operates without electrical energy and less water consumption. Fully bleached pine fiber (Softwood kraft pulp) pretreated with 15% NaOH w/v was used as the raw material. Wet-laid nonwoven fabric of 200 g/m2, 300 g/m2, and 400 g/m2 mass per unit area were produced and tested for tensile strength and elongation were found to confirm the integrity of the nonwoven. Nonwoven fabric air permeability decreased with increasing mass per unit area and amount 38.08 ± 1.57 cm3/s/cm2, 29.75 ±0.49 cm3/s/cm2, 22.86 ± 0.88 cm3/s/cm2. The mean pore size was 36.42 µm, 33.66 µm, and 32.37 µm and the total absorption capacity was 20.57 ± 1.3 g/g, 30.05 ± 2.1 g/g, and 36.02 ± 0.88 g/g, respectively. SEM image shows random arrangement and interlacement of fibers and the nonwoven produced was found to be smooth, soft, and flexible using tissue softness analyser. Thus, the designed wet-laid equipment can be used to produce cost-effective wet-laid nonwovens and it is an eco-friendly setup regarding water recycling and the produced nonwoven samples are suitable for the absorbent core of the disposable nursing pad.

An Improved Decoupling Inequality for Random Interlacements

In this paper we obtain a decoupling feature of the random interlacements process $\mathcal{I}^u \subset \mathbb{Z}^d$, at level $u$, $d\geq 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $\textsf{F}$ and its translated $\textsf{F}+x$, can be coupled with high probability of success, when $\|x\|$ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two $[0,1]$-valued functions depending on the configuration of the random interlacements on $\textsf{F}$ and $\textsf{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].

On macroscopic holes in some supercritical strongly dependent percolation models

We consider $Z^d$, with d bigger or equal to three. We investigate the vacant set of random interlacements in the strongly percolative regime, the vacant set of the simple random walk, and the excursion set above a given level of the Gaussian free field in the strongly percolative regime. We derive asymptotic upper and lower exponential bounds for the large deviation probability that the adequately thickened component of the boundary of a large box centered at the origin in the respective vacant sets or excursion set leaves in the box a macroscopic volume in its complement. We also derive geometric information on the shape of the left-out volume. It is plausible, but open at the moment, that certain critical levels coincide, both in the case of random interlacements and of the Gaussian free field. If this holds true, the asymptotic upper and lower bounds that we obtain are matching in principal order for all three models, and the macroscopic holes are nearly spherical. We heavily rely on the recent work arXiv:1706.07229 by Maximilian Nitzschner and the author for the coarse graining procedure, which we employ in the derivation of the upper bounds.

Two-Dimensional Brownian Random Interlacements

We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, 2016], as well as the results specific to the continuous case.

Finitary random interlacements and the Gaboriau–Lyons problem

The von Neumann–Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol’shanskii in the 1980s. The measurable version (formulated by Gaboriau–Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau–Lyons. The proof uses an approximation to the random interlacement process by random multisets of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau–Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.

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.

On coupling and “vacant set level set” percolation

In this note we discuss “vacant set level set” percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of ${\mathbb Z}^d$, $d \ge 3$.

How can a clairvoyant particle escape the exclusion process?

We study a detection problem in the following setting: On the one-dimensional integer lattice, at time zero, place nodes on each site independently with probability $\rho \in [0,1)$ and let them evolve as a simple symmetric exclusion process. At time zero, place a target at the origin. The target moves only at integer times, and can move to any site that is within distance $R$ from its current position. Assume also that the target can predict the future movement of all nodes. We prove that, for $R$ large enough (depending on the value of $\rho$) it is possible for the target to avoid detection forever with positive probability. The proof of this result uses two ingredients of independent interest. First we establish a renormalisation scheme that can be used to prove percolation for dependent oriented models under a certain decoupling condition. This result is general and does not rely on the specifities of the model. As an application, we prove our main theorem for different dynamics, such as independent random walks and independent renewal chains. We also proof existence of oriented percolation for random interlacements and for its vacant set for large dimensions . The second step of the proof is a space-time decoupling for the exclusion process.

The Sign Clusters of the Massless Gaussian Free Field Percolate on $${\mathbb{Z}^{d}, d \geqslant 3}$$ Z d , d ⩾ 3 (and more)

We investigate the percolation phase transition for level sets of the Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $d\geqslant3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $\mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $\mathbb{Z}^d$.

On pinned fields, interlacements, and random walk on $$({\mathbb {Z}}/N {\mathbb {Z}})^2$$ ( Z / N Z ) 2

We define two families of Poissonian soups of bidirectional trajectories on $${\mathbb {Z}}^2$$ , which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus $$({\mathbb {Z}}/N {\mathbb {Z}})^2$$ , started from the uniform distribution, run up to a time of order $$(N\log N)^2$$ and forced to avoid a fixed point. The local limit of the latter was recently established in Comets et al. (Commun Math Phys 343:129–164, 2016). Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of “finite volume” approximations, consisting of random walks avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity u of such walks with N, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced in Sznitman (Ann Math 171(3):2039–2087, 2010) in the transient case. It also links it to the pinned free field in $${\mathbb {Z}}^2$$ , by means of a (pinned) Ray–Knight type isomorphism theorem.

Interlacements and the wired uniform spanning forest

We extend the Aldous–Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman’s random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of ‘excessive ends’ in the WUSF is nonrandom in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm [Electron. J. Probab. 13 (2008) 1702–1725], while the third extends a recent result of the author. Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.

Level-set percolation for the Gaussian free field on a transient tree

We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_*$ and $u_*$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_* < \sqrt{2u}_*$ in a broad enough set-up, but provide an example where $0 = h_* = u_*$ occurs. We also obtain some sufficient conditions ensuring that $h_* > 0$.