Let K and K₀ be convex bodies in R^d, such that K contains the origin, and define the process (K_n,p_n), n≥0, as follows: let p_n+1 be a uniform random point in K_n, and set K_n+1 = K_n ∩ (p_n+1 + K). Clearly, (K_n) is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in R^d. First, we investigate the process in one dimension, when K=[-1,1]. In this case the limit set is a unit interval, and the distribution of its center has the arcsine law. Further, we study this process for K being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices.
This is joint work with Gergely Ambrus and Viktor Vigh.

We derive an error exponent for frame-asynchronous discrete memoryless multiple access system with two senders. Moreover, our coding/decoding scheme is universal. This have been done by extending Csiszár-Körner-Marton's packing lemma to asynchronous system and using a modified "maximal multi-information" decoder. This aforementioned extension is quite straightforward, employing the new concept of sub-types introduced by us. This joint result with Tamás Kói will be presented at the annual Information Theory Symposium (ISIT 2014).
Some new corollaries will be presented:

• The universal exponent is strictly positive inside the capacity region. This admits us to deal with the compound asynchronous model.
• The new concept of "Controlled Asynchronism" is defined. In some cases, this gives a better exponent than the synchronous counterpart (we present an example).
• It seems a new single user detection algorithm is defined, which can be used to detect multiple access system.

We address a version of the random access model of J. Luo and A. Ephremides (supplemented by Z. Wang and J. Luo), which is similar to a model studied for one-way channels by I. Csiszar. The results of I. Csiszar are generalized to (discrete memoryless) multiple access channels (MACs). A two-senders random access model is introduced, in which the senders have codebook libraries with constant composition codebooks for multiple rate choices. The error exponent of Y. Liu and B.L. Hughes for an individual codebook pair is shown to be simultaneously achievable for each codebook pair in the codebook libraries, supplemented with collision detection. This is achieved via universal decoder. Moreover, achievable joint source-channel coding error exponents for transmitting independent sources over a MAC are given, admitting improvements when error free 0 rate communication is allowed between the two senders. The most direct extension of the results of I. Csiszar is obtained in the latter case.
The work is joint with Lorant Farkas. Manuscript can be downloaded from http://arxiv.org/pdf/1301.6412v3.pdf

I will describe the Spatial Lambda-Fleming-Viot process, which is a recent model of evolution in a spatial continuum, and explore the time and spatial scale on which selectively advantageous genes propagate through space. The appropriate scaling depends on the dimension of space, resulting in three distinct cases (d=1, d=2 and d>=3). In d=1 I will outline why, in this case, the limiting genealogy is the Brownian net.

The mean field forest fire model was introduced by Ráth and Tóth in 2009. The model is a continuous-time Markov process on the space of graphs on n labelled vertices. As in the dynamical Erdős-Rényi model, edges appear independently at rate 1/n per edge, with the result that increasingly large clusters of connected vertices form. Independently, each vertex is struck by lightning at some rate depending on n; when this happens all the edges in the cluster of that vertex instantaneously disappear. Ráth and Tóth showed that for suitable values of the lightning rate, the system exhibits self-organized criticality in the large n limit.
Building on their work, we study the local limit of the model, understanding how the process appears from the point of view of a single vertex, as the size of the model tends to infinity. The limit can be understood as an explosive Markov branching process with a deterministically varying offspring distribution. The environment evolves according to an infinite system of ODEs called the critical forest fire equations, which are a simple modification of the well-known Smoluchowski coagulation equations with multiplicative kernel.
Finally we explain how the Brownian continuum random tree arises as the scaling limit of the large clusters. (Joint work with Nic Freeman and Bálint Tóth)

It is an open problem whether the limiting eigenvalue distribution of percolation in a large box has an absolutely continuous part. Similarly, it is not known whether the analogue of the Wigner semicircle law for a sparse Erdos-Renyi random matrix has an absolutely continuous part. In joint work with Charles Bordenave and Arnab Sen we can show that non-atomic part exists in many models. I will review the background and some new methods.

We consider the contact process (SIS epidemic) on finite graphs. Unlike infinite graphs, which can exhibit a phase transition between the process surviving forever and becoming extinct, the process always dies out on finite graphs. Nevertheless, there is a signature of the phase transition in that the process lifetime can show a sharp change on large graphs, between being logarithmic and exponential in the number of nodes.In this talk, we will give necessary conditions for these two regimes, of short-lived and long-lived contact processes, in terms of graph properties, namely the spectral radius and the isoperimetric constant. We will also discuss when these match, implying a sharp threshold between these regimes. We will extend the results to some commonly used random graph models. Finally, we will discuss an application of these ideas to the optimal allocation of "curing resources" in an epidemic.

I will present several models of random geometric subdivisions, similar to that of Diaconis and Miclo (Combinatorics, Probability and Computing, 2011), where a triangle is split into 6 smaller triangles by its medians, and one of these parts is randomly selected as a new triangle, and the process continues ad infinitum. I will show that in a similar model the limiting shape of an indefinite subdivision of a quadrilateral is a parallelogram. I will also show that the geometric subdivisions of a triangle by angle bisectors converge (but only weakly) to a non-atomic distribution, and that the geometric subdivisions of a triangle by choosing a uniform random points on its sides converges to a flat  triangle, similarly to the result of the paper mentioned above.

Egy gráf csúcsainak egy halmaza független, ha a halmaz semelyik két csúcsa között nem megy él. Egy természetes és fontos gráfparaméter az ilyen független halmazok maximális mérete egy adott gráfban, ezt a gráf függetlenségi számának nevezzük. Véletlen reguláris gráfokban először Bollobás adott felső becslést a függetlenségi számra. Azóta számos fejlemény történt, ebben az előadásban elsősorban lokális algoritmussal megkonstruálható (véletlen) független halmazokról lesz szó. Egy új megközelítés a gráf minimális sajátértékét és az ahhoz tartozó véletlen sajátvektorokat használ. Az elődás Csóka Endrével, Gerencsér Balázssal, Virág Bálinttal közös munkákon alapul.

Egy gráf csúcsainak egy halmaza független, ha a halmaz semelyik két csúcsa között nem megy él. Egy természetes és fontos gráfparaméter az ilyen független halmazok maximális mérete egy adott gráfban, ezt a gráf függetlenségi számának nevezzük. Véletlen reguláris gráfokban először Bollobás adott felső becslést a függetlenségi számra. Azóta számos fejlemény történt, ebben az előadásban elsősorban lokális algoritmussal megkonstruálható (véletlen) független halmazokról lesz szó. Egy új megközelítés a gráf minimális sajátértékét és az ahhoz tartozó véletlen sajátvektorokat használ. Az elődás Csóka Endrével, Gerencsér Balázssal, Virág Bálinttal közös munkákon alapul.