Syllabus:
What is a bidimensional random geometry ? A possible definition is
provided by maps, i.e. graphs embedded in surfaces and considered up to (...)
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What is a bidimensional random geometry ? A possible definition is
provided by maps, i.e. graphs embedded in surfaces and considered up to
deformation. These objects received a lot of attention in combinatorics,
theoretical physics and more recently in probability theory. In this
workgroup, we will be particularly interested in local limits: the
notion of local convergence of a family of graphs was introduced by
Benjamini and Schramm [1], and Angel and Schramm showed that uniform
planar triangulations converge in this sense [2]. The question of
whether the simple random walk on the limiting object is recurrent has
remained open for a long time, it was recently settled by Gurel-Gurevich
and Nachmias [3]. We will also study other properties of local limits
such as percolation thresholds [4].
[1] Itai Benjamini and Oded Schramm. Recurrence of Distributional
Limits of Finite Planar Graphs. Electron. J. Probab. 6 (2001), no. 23,
13 pp. (electronic).
[2] Omer Angel and Oded Schramm. Uniform infinite planar
triangulations. Comm. Math. Phys. 241 (2003), no. 2-3, 191–213.
[3] Ori Gurel-Gurevich and Asaf Nachmias. Recurrence of planar graph
limits. Ann. of Math. (2) 177 (2013), no. 2, 761–781.
[4] Omer Angel and Nicolas Curien. Percolations on random maps I:
Half-plane models. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015),
no. 2, 405–431.
