arXiv subjects

Alain-Sol Sznitman

Publications and source records attributed to Alain-Sol Sznitman.

33 records · Page 2Linked to original sources

arXiv2013

Phase transition and level-set percolation for the Gaussian free field

We consider level-set percolation for the Gaussian free field on Z^d, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h > h_*, the excursion set above level h does not percolate, and for h < h_*, the excursion set does percolate. It is known from the work of Bricmont-Lebowitz-Maes that h_* is non-negative for all d bigger or equal to 3, and finite, when d=3. We prove here that h_* is finite for all d bigger or equal to 3. In fact, we introduce a second critical parameter h_**, which is bigger or equal to h_*. We show that h_** is finite for all d bigger or equal to 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h_**. Finally we prove that h_* > 0 in high dimension. It remains open whether h_* and h_** actually coincide, and whether h_* > 0 for all d bigger or equal to 3.

math.PR↗

arXiv2012

Random interlacements and the Gaussian free field

We consider continuous time random interlacements on $\mathbb{Z}^d$, $d \ge 3$, and characterize the distribution of the corresponding stationary random field of occupation times. When d = 3, we relate this random field to the two-dimensional Gaussian free field pinned at the origin by looking at scaled differences of occupation times of long rods by random interlacements at appropriately tuned levels. In the main asymptotic regime, a scaling factor appears in the limit, which is independent of the free field, and distributed as the time-marginal of a zero-dimensional Bessel process. For arbitrary $d \ge 3$, we also relate the field of occupation times at a level tending to infinity, to the d-dimensional Gaussian free field.

math.PR↗

arXiv2011

An isomorphism theorem for random interlacements

We consider continuous-time random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second Ray-Knight theorem, and uniquely determines the law of occupation times of random interlacements at level u.

math.PR↗

arXiv2011

Decoupling inequalities and interlacement percolation on G x Z

We study the percolative properties of random interlacements on the product of G with the integer line Z, when G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters alpha > 1, measuring the volume growth on G, and beta between 2 and alpha + 1, measuring the sub-diffusive nature of the random walk on G. We develop decoupling inequalities, which are a key tool in showing that the critical level u_* for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when alpha \geq 1 + beta/2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where G = Z^d, d \geq 2, several of these results are new.

math.PR↗

arXiv2010

On the critical parameter of interlacement percolation in high dimension

The vacant set of random interlacements on ${\mathbb{Z}}^d$, $d\ge3$, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039--2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831--858] that there is a nondegenerate critical value $u_*$ such that the vacant set at level $u$ percolates when $u u_*$. We derive here an asymptotic upper bound on $u_*$, as $d$ goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that $u_*$ is equivalent to $\log d$ for large $d$ and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on $2d$-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604--1627].

math.PR↗

arXiv2010

A lower bound on the critical parameter of interlacement percolation in high dimension

We investigate the percolative properties of the vacant set left by random interlacements on Z^d, when d is large. A non-negative parameter u controls the density of random interlacements on Z^d. It is known from arXiv:0704.2560, and arXiv:0808.3344, that there is a non-degenerate critical value u_*, such that the vacant set at level u percolates when u < u_*, and does not percolate when u > u_*. Little is known about u_*, however for large d, random interlacements on Z^d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, for which the corresponding critical parameter can be explicitly computed, see arXiv:0907.0316. We prove in this article a lower bound on u_*, which is equivalent to log(d) as d goes to infinity. This lower bound is in agreement with the above mentioned heuristics.

math.PR↗

arXiv2009

Vacant Set of Random Interlacements and Percolation

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder with base a d-1-dimensional discrete torus of side-length N, or the set of points visited by simple random walk on the d-dimensional discrete torus of side-length N by times of order uN^d. We study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite, connected, translation invariant random subset of Z^d. We introduce a critical value such that the vacant set percolates for u below the critical value, and does not percolate for u above the critical value. Our main results show that the critical value is finite when d is bigger or equal to 3, and strictly positive when d is bigger or equal to 7.

math.PR↗

arXiv2009

Upper bound on the disconnection time of discrete cylinders and random interlacements

We study the asymptotic behavior for large $N$ of the disconnection time $T_N$ of a simple random walk on the discrete cylinder $(\mathbb{Z}/N\mathbb{Z})^d\times\mathbb{Z}$, when $d\ge2$. We explore its connection with the model of random interlacements on $\mathbb{Z}^{d+1}$ recently introduced in [Ann. Math., in press], and specifically with the percolative properties of the vacant set left by random interlacements. As an application we show that in the large $N$ limit the tail of $T_N/N^{2d}$ is dominated by the tail of the first time when the supremum over the space variable of the Brownian local times reaches a certain critical value. As a by-product, we prove the tightness of the laws of $T_N/N^{2d}$, when $d\ge2$.

math.PR↗

arXiv2009

Connectivity Bounds for the Vacant Set of Random Interlacements

The model of random interlacements on Z^d, d bigger or equal to 3, was recently introduced in arXiv:0704.2560. A non-negative parameter u parametrizes the density of random interlacements on Z^d. In the present note we investigate the connectivity properties of the vacant set left by random interlacements at level u, in the non-percolative regime, where u is bigger than the non-degenerate critical parameter for percolation of the vacant set, see arXiv:0704.2560, arXiv:0808.3344. We prove a stretched exponential decay of the connectivity function for the vacant set at level u, when u is bigger than an other critical parameter. It is presently an open problem whether these two critical parameters actually coincide.

math.PR↗

arXiv2009

On the Domination of Random Walk on a Discrete Cylinder by Random Interlacements

We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order N, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite (d+1)-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when d is at least 17, in arXiv: math/0701414.

math.PR↗

arXiv2008

Percolation for the Vacant Set of Random Interlacements

We investigate random interlacements on Z^d, d bigger or equal to 3. This model recently introduced in arXiv:0704.2560 corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time-shift tending to infinity at positive and negative infinite times. A non-negative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d bigger or equal to 3 the vacant set at level u percolates when u is small. This solves an open problem of arXiv:0704.2560, where this fact has only been established when d is bigger or equal to 7. It also completes the proof of the non-degeneracy in all dimensions d bigger or equal to 3 of the critical parameter introduced in arXiv:0704.2560.

math.PR↗

arXiv2008

A Lower Bound on the Disconnection Time of a Discrete Cylinder

We study the asymptotic behavior for large N of the disconnection time T_N of simple random walk on a discrete cylinder with base a d-dimensional discrete torus of side-length N. When d is sufficiently large, we are able to substantially improve the lower bounds obtained by the authors in a previous article when d is bigger or equal to 2. We show here that the laws of N^(2d)/T_N are tight.

math.PR↗

arXiv2008

Random Walks on Discrete Cylinders and Random Interlacements

We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of random interlacements recently introduced in arXiv:0704.2560. In particular we show that when the base becomes large, in the neighborhood of a point of the cylinder with a vertical component of order N^d, the complement of the set of points visited by the walk up to times of order N^{2d}, is close in distribution to the law of the vacant set of random interlacements at a level which is determined by an independent Brownian local time. The limit of the local pictures in the neighborhood of finitely many points is also derived.

math.PR↗

arXiv2008

Giant Component and Vacant Set for Random Walk on a Discrete Torus

We consider random walk on a discrete torus E of side-length N, in sufficiently high dimension d. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uN^d. We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const log N. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites N^d of E, and any point of E lies within distance an arbitrary fractional power of N from this component.

math.PR↗

arXiv2007

How universal are asymptotics of disconnection times in discrete cylinders?

We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large $N$ the disconnection time of $G_N\times\mathbb{Z}$ has rough order $|G_N|^2$, when $G_N=(\mathbb{Z}/N\mathbb{Z})^d$. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree.

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