601 research outputs found

### Exceptional times for percolation under exclusion dynamics

We analyse in this paper a conservative analogue of the celebrated model of
dynamical percolation introduced by H\"aggstr\"om, Peres and Steif in [HPS97].
It is simply defined as follows: start with an initial percolation
configuration $\omega(t=0)$. Let this configuration evolve in time according to
a simple exclusion process with symmetric kernel $K(x,y)$. We start with a
general investigation (following [HPS97]) of this dynamical process $t \mapsto
\omega_K(t)$ which we call $K$-exclusion dynamical percolation. We then proceed
with a detailed analysis of the planar case at the critical point (both for the
triangular grid and the square lattice $Z^2$) where we consider the power-law
kernels $K^\alpha$ $K^{\alpha}(x,y) \propto \frac 1 {\|x-y\|_2^{2+\alpha}} \,
.$ We prove that if $\alpha > 0$ is chosen small enough, there exist
exceptional times $t$ for which an infinite cluster appears in
$\omega_{K^{\alpha}}(t)$. (On the triangular grid, we prove that it holds for
all $\alpha < \alpha_0 = \frac {217}{816}$.) The existence of such exceptional
times for standard i.i.d. dynamical percolation (where sites evolve according
to independent Poisson point processes) goes back to the work by Schramm-Steif
in [SS10]. In order to handle such a $K$-exclusion dynamics, we push further
the spectral analysis of exclusion noise sensitivity which had been initiated
in [BGS13]. (The latter paper can be viewed as a conservative analogue of the
seminal paper by Benjamini-Kalai-Schramm [BKS99] on i.i.d. noise sensitivity.)
The case of a nearest-neighbour simple exclusion process, corresponding to the
limiting case $\alpha = +\infty$, is left widely open.Comment: 50 pages, 6 figures, there was a problem with the compilation of the
tex fil

### On the convergence of FK-Ising Percolation to SLE$(16/3, 16/3-6)$

We give a simplified and complete proof of the convergence of the chordal
exploration process in critical FK-Ising percolation to chordal SLE$_\kappa(
\kappa-6)$ with $\kappa=16/3$. Our proof follows the classical
excursion-construction of SLE$_\kappa(\kappa-6)$ processes in the continuum and
we are thus led to introduce suitable cut-off stopping times in order to
analyse the behaviour of the driving function of the discrete system when
Dobrushin boundary condition collapses to a single point. Our proof is very
different from [KS15, KS16] as it only relies on the convergence to the chordal
SLE$_{\kappa}$ process in Dobrushin boundary condition and does not require the
introduction of a new observable. Still, it relies crucially on several
ingredients:
a) the powerful topological framework developed in [KS17] as well as its
follow-up paper [CDCH$^+$14],
b) the strong RSW Theorem from [CDCH16],
c) the proof is inspired from the appendix A in [BH16].
One important emphasis of this paper is to carefully write down some
properties which are often considered {\em folklore} in the literature but
which are only justified so far by hand-waving arguments. The main examples of
these are:
1) the convergence of natural discrete stopping times to their continuous
analogues. (The usual hand-waving argument destroys the spatial Markov
property).
2) the fact that the discrete spatial Markov property is preserved in the the
scaling limit. (The enemy being that $\mathbb{E}[X_n |\, Y_n]$ does not
necessarily converge to $\mathbb{E}[X|\, Y]$ when $(X_n,Y_n)\to (X,Y)$).
We end the paper with a detailed sketch of the convergence to radial
SLE$_\kappa( \kappa-6)$ when $\kappa=16/3$ as well as the derivation of
Onsager's one-arm exponent $1/8$.Comment: 35 pages, 7 figures. Final version, to appear in Journal of
Theoretical Probabilit

### The expected area of the filled planar Brownian loop is Pi/5

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion
conditioned so that B_0 = B_1). We consider the compact hull obtained by
filling in all the holes, i.e. the complement of the unique unbounded component
of R^2\B[0,1]. We show that the expected area of this hull is Pi/5. The proof
uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also,
using the result of Yor about the law of the index of a Brownian loop, we show
that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2).
As a consequence, we find that the expected area of the region of index zero
inside the loop is Pi/30; this value could not be obtained directly using Yor's
index description.Comment: 15 pages, 3 figure

### The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio

### Coalescing Brownian flows: A new approach

The coalescing Brownian flow on $\mathbb{R}$ is a process which was
introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ.
Wisconsin, Madison] and T\'{o}th and Werner [Probab. Theory Related Fields 111
(1998) 375-452], and which formally corresponds to starting coalescing Brownian
motions from every space-time point. We provide a new state space and topology
for this process and obtain an invariance principle for coalescing random
walks. This result holds under a finite variance assumption and is thus
optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883],
Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and
state-space required a moment of order $3-\varepsilon$ for this convergence to
hold. The proof relies crucially on recent work of Schramm and Smirnov on
scaling limits of critical percolation in the plane. Our approach is
sufficiently simple that we can handle substantially more complicated
coalescing flows with little extra work - in particular similar results are
obtained in the case of coalescing Brownian motions on the Sierpinski gasket.
This is the first such result where the limiting paths do not enjoy the
noncrossing property.Comment: Published at http://dx.doi.org/10.1214/14-AOP957 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org

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