Uniform infinite planar triangulation and related time - reversed critical branching process ∗

We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e. for a boundary component separating the ball from infinity). We show also that outside of R-ball a contour exists that has length linear in R. Introduction The uniform infinite planar triangulation (UIPT) is a random graph, considered as one of possible models of generic planar geometry. UIPT is defined as a weak limit of uniform measures on triangulations with finite number of triangles. In [1] Angel ans Schramm proved the existence of this limit, in [2] some basic geometrical properties of UIPT were investigated, in particular it was found that the ball of radius R in UIPT has volume of order R and boundary of order R, up to polylogarithmic terms. This fact reflects the conjecture known in physics, see [7]. In this paper we improve the result of [2] concerning the boundary of a ball and give an exact limit of a corresponding scaled random variable asR → ∞. We use a new combinatorial ”skeleton” construction, which uncovers a connection between UIPT profile and certain time-reversed branching process. Using this connection we state a new fact concerning the UIPT: we show that outside of the R-ball a contour exists, that separates the ball from the infinite part of triangulation and has length linear in R. The paper is organized as follows. In the first part we give necessary definitions and review some results of [1, 2]. The main results of this paper are stated in section 1.4. In the second part we describe the skeleton construction. In the third part we use the ”raw method” to obtain the limiting distribution of This research was partially supported by RFBR grant 02-01-00415. Laboratory of Large Random Systems, Faculty of Mechanics and Mathematics, Moscow State University. E-mail: [email protected]