Random Geometry.

Tuesday 24 January 2017, IHP

Moderator: Bertrand Duplantier (Saclay)

Juhan Aru (Zürich), Gaëtan Borot (Bonn), Emmanuel Guitter (Saclay), Fredrik Johansson Viklund (Stockholm), Jean-François Le Gall (Orsay), Jason Miller (Cambridge), Hao Wu (Genève & Beijing)

Juhan Aru (Zürich): Unit area quantum sphere.

I will discuss different probabilistic perspectives on the so called unit area quantum sphere (also sometimes called the unit volume Liouville measure), appearing in the works of Duplantier, Miller & Sheffield and David, Kupiainen, Rhodes & Vargas.

Gaëtan Borot (Bonn): Nesting of loops on random surfaces.

I will explain how analytic combinatorics allows the enumeration of surfaces carrying a configuration of the O(n) loop model, as well as tracking separating vs. non-separating loops. In particular, we compute the distribution number of separating loops between two marked elements in a planar random map, and its resulting large deviations function at criticality, which is universal. This large deviation compares well, by an application of KPZ transformation, to the large deviation of the number of separating loops in a sample of CLE_{k} with n = 2cos pi|1 - 4/k| on a planar domain, observed with an independent sample of the area measure of Liouville quantum gravity. This provides extra support to the general 2d quantum gravity conjecture.
Based on joint works with Emmanuel Guitter, Jeremie Bouttier, Bertrand Duplantier, and Elba Garcia-Failde.

Emmanuel Guitter (Saclay): Universal laws for hulls in large planar maps. pdf

Consider an ensemble of planar maps, say planar quadrangulations, with a fixed number N of faces and with two marked vertices at distance k from each other. Consider then, for d < k, the closed line separating these vertices and lying at distance d from the first one. This line divides the map into two complementary components, the component containing the first vertex being called the "hull at distance d". This hull is charaterized by its volume (number of faces) and its perimeter, which is the length (number of edges) of the separating line itself. We will discuss the statistics of hull volumes and perimeters in the limit where N is infinite and d and k become large with a fixed ratio d/k. Two regimes are encountered: either the hull has a finite volume and its complementary is of infinite size, or the hull itself is infinitely large and its complementary remains finite. We will compute the probability for the map to be in either regime as a function of d/k and present explicit universal laws for the statistics of hull perimeters and volumes when maps are conditioned to be in one regime or the other. These results follow from a coding of maps by slices together with a recursive decomposition of these slices obtained by cutting them along separating lines at increasing distances from the first marked vertex.

Fredrik Johansson Viklund (Stockholm): Convergence of loop-erased walk in the natural parametrization.

Loop-erased random walk (LERW) is the random self-avoiding path one gets after erasing the loops in the order they form from a simple random walk or alternatively by considering the branches in a uniform spanning tree. Lawler, Schramm and Werner proved that LERW in 2D converges to SLE(2) as curves viewed up to reparametrization. It is however more natural to view the discrete curve parametrized by (renormalized) length and it has been believed for some time that one then has convergence to SLE(2) equipped with the so-called natural parametrization, which in this case is the same as 5/4-dimensional Minkowski content. I will discuss recent joint works with Greg Lawler (Chicago) that prove this stronger convergence, focusing on explaining the main ideas of the argument.

Jean-François Le Gall (Orsay): First-passage percolation in random planar lattices.

We consider local modifications of the graph distance in random graphs drawn in the plane, which are also called random planar maps. A particular case is the first-passage percolation distance, where edges of the graph are assigned i.i.d. random lengths. We show that in large scales the associated first-passage percolation distance is asymptotically proportional to the graph distance.
For the infinite random lattice known as the UIPT (uniform infinite planar triangulation), this means that large balls for the first-passage percolation distance behave asymptotically like deterministic balls. This is in sharp contrast with the results known or conjectured to hold for deterministic lattices. In the special case corresponding to the so-called Eden model on random triangulations, we are able to compute the explicit value of the mutiplicative constant.
This talk is based on a joint work with Nicolas Curien.

Jason Miller (Cambridge): Convergence of the self-avoiding walk on random quadrangulations to SLE_{8/3} on \sqrt{8/3}-Liouville quantum gravity.

Let (Q,\lambda) be a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk. We prove that (Q,lambda) converges in the scaling limit to a certain \sqrt{8/3}-Liouville quantum gravity surface decorated by an independent chordal SLE_{8/3}. The scaling limit can equivalently be described as the metric gluing of two independent instances of the Brownian half-plane. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the Gromov-Hausdorff topology to curve-decorated metric measure spaces.
Based on joint work with E. Gwynne.

Hao Wu (Genève & Beijing): Arm Exponents for Ising and FK-Ising Model. pdf

The introduction of SLE by Oded Schramm provides mathematicians with a new tool to study critical lattice models. In the first part of this talk, I will discuss critical planar percolation and explain how people use SLE to derive the arm exponents of percolation. In the second part, I will introduce SLE and general formulae on arm exponents of SLE, and show that these formulae give us various results on the arm exponents of critical planar Ising and FK-Ising models. Finally, I will explain some related results and open questions.