Research interests

Random surfaces

In the last few years several researchers, including Schaeffer, Schramm and Angel, developped some models of random planar maps, which differ from traditional random graphs in that there is no underlying lattice (such az Zd) or any "canonical" embedding into the Euclidean space.

These surfaces are considered as a model for a random metric -- one puts all triangles or squares equilatteral, thus the surface is almoust flat, and teh curvature is discrete and is concentrated in the vertices.

The models considered were proved to have very stange geometry:

• the "disk" of radius R on such surface has boundary length of order R2 and surface square of order R4;
• the diameter of a map that consists of N4 squares has order N{1/4} In other words, the internal dimension of such a surface is 4.

Problems

Continous limit

The open question is wheter there exists certain continous limit of these discrete random surfaces under certain scaling.

There are now at least two bridges from "x-angulations" to "brownian-

something

":

• Schaeffer showed that the random quadrangulations lead to the integrated superbrownian excursion.
• On the other hand, the triangulations are related to time-reversed critical branching processes. The critical branching processes in their turn are known to converge (in certain sense) to Aldous's continuum random tree, which is closely related to the brownian excursion.

Some elements of the last chain are still missing. I hope it will be completed soon.

Dynamics

There is a demand for a dynamical model of infinte random triangulation. In particular one would find a dynamics that has uniform measure it's stable measure.

One way to define such a dynamics is through flips and splits technique. Unfortunately this method is not closed (in certain sence); the result of elementary transforms considered may lay in the same triangulation class as the initial triangulation (see TriangulationTypes).

As a quick fix to this disadvantage of flips and splits is a non-local dynamics, when one is allowed to add or remove certain non-trivial parts of triangulation at one step.

Geometry

To better understand the nature of a supposed limit, one can translate some notions from differential geometry to a random maps model and see what happens. What other natural geometrical notions could be considered?

• Circle length. The growth theorem, saying that the boundary of an R-hull grows asymptoticaly as R2, is an analog of circle length measurement. The length of a circle with given radius r for planar or Euclidean geometry is equal, for spherical geometry is less and for hyperbolic geometry is greater than 2π r.
• Geodesic lines. In a random triangulation a geodesic line can be defined in a purely combinatorial way, that has however geometric sence. Such lines may be closed, self-intersecting. An interesring question is whether there are infinite geodesics, and what is the probability for this.
• Curvature. In fact a discrete triangulation can be considered as a manifold with singular curvature concentrated at vertices.
• What else?

Related publications:

(in fact these are not the publications on random surfaces, but the papers that could be applied to the chain above)

• D.Aldous, Triangulating the Circle, at Random. Amer. Math. Monthly 101 (1994) 223-233.
• Dr. Kersting, On the height profile of a conditioned Galton-Watson tree, preprint (1998) http://ismi.math.uni-frankfurt.de/kersting/research/profile.ps
• Drmota, Gittenberger, On the profile of random trees, Random Structures and Algorithms 10, 421-451, 1997
• Drmota, Gittenberger, The width of Galton-Watson trees, submitted
• Drmota, Gittenberger, Strata of random mappings - a combinatorial approach, Stochastic Processes and their Applications 82, 157-171, 1999.

Radnom Trees and Random Grammars

A research on a dynamical random tree model, that resulted in a 2004 paper with G.Fayolle and J-M.Lasgouttes, see (PreprintsAndPublications), is in fact related to a certain random grammar (context-dependent).

A well-known depth-first search algorithm gives a bijection between planar trees and parenthesis structures, or words in a 2-letter alphabet

U, D

. Under this bijection the markov process on trees becomes a random grammar process on words. Namely, there are two transitions:
U → UDU, with rate λ


DU → ε, with rate μ

(here ε stands for the empty word). In order to keep the root of the tree immortal, one should start the process from the word "D".

It would be interesting to consider the methods used to study the dynamical random tree model in application to some more general random grammar.

There are few other problems, that are not solved yet, in particular there should exist some limit for the shape of the tree apex (top) in the transient case.

Other interests

• MathAndProgramming