# Sur le papier "RIGIDITY AND TOLERANCE FOR PERTURBED LATTICES" de Sly et Peres

## Géométrie Stochastique

Lieu:
Salle séminaire M3-324
Orateur:
David Dereudre
Dates:
Mercredi, 28 Novembre, 2018 - 14:00 - 16:00
Résumé:
A perturbed lattice is a point process Π = {x+Y x : x ∈ Z d }
where the lattice points in Zd are perturbed by i.i.d. random variables
{Yx } x∈Zd . A random point process Π is said to be rigid if |Π∩B 0(1)|, the
number of points in a ball, can be exactly determined given Π \ B 0(1),
the points outside the ball. The process Π is called deletion tolerant if
removing one point of Π yields a process with distribution indistinguish-
able from that of Π. Suppose that Yx ∼ Nd (0, σ2I) are Gaussian vectors
with with d independent components of variance σ2 . Holroyd and Soo
showed that in dimensions d = 1, 2 the resulting Gaussian perturbed lat-
tice Π is rigid and deletion intolerant. We show that in dimension d ≥ 3
there exists a critical parameter σr(d) such that Π is rigid if σ < σ r and
deletion tolerant (hence non-rigid) if σ > σr.