We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. The strategy behind the large deviation event is based on a condensation effect for the vertex degrees. A finite number of vertices are selected at random and their power increased so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We also show that the empirical distribution of edge lengths under the conditioning splits into a bulk and travelling wave part of asymptotically positive proportions. The talk is based on a collaboration with Pim van der Hoorn, Remco van der Hofstad, Céline Kerriou and Neeladri Maitra.

In this talk I will present the local limit of preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random Pólya point tree, has a surprising size-biasing phenomenon. Our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. We shall use the local limit to identify the percolation phase transition on preferential attachment models. We identify the critical percolation threshold explicitly using the branching process limit. This is based on joint works with Alessandro Garavaglia, Remco van der Hofstad and Rounak Ray.

We consider a general class of branching Markov processes whose offspring distribution is non-local. In the supercritical regime, the first order long-term behaviour is well understood via Perron Frobenius decompositions, laws of large numbers and the asymptotic behaviour of its moments. In this talk, we consider the second order fluctuations of the process in three different regimes, depending on the sign of the spectral gap. In each regime, we prove a functional limit theorem that characterises the long-term fluctuations of the process.

We study a d-dimensional random walk with independent and identically  distributed increments in a Weyl chamber.  We discuss existence of the harmonic function for the random walks killed on leaving the Weyl chamber . We then construct an ordered process using Doob's h-transform. For this process we discuss limit theorems when d is fixed and when d slowly increases.

In this talk, we main explore the macroevolution mechanism of local times of a spectrally positive stable process in the spatial direction. The main results state that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, the local times at positive half line are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson random measure whose intensity coincides with the L\’evy measure. This helps us to provide not only a simple proof for the H\”older regularity, but also a uniform upper bound for all moments of the H\”older coefficient as well as a maximal inequality for the local times. Moreover, based on this stochastic Volterra equation, we extend the method of duality to establish an exponential-affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra integral equation associated with the Laplace exponent of the stable process.  

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A random walk in sparse random environment is a model in which the particle performs a simple random walk on the set of integers. It moves symmetrically except for some randomly chosen sites, where random drift is imposed. During my talk I will present some properties of the model and the results on quenched limit laws obtained recently with D. Buraczewski and P. Dyszewski 

Let $(g_n)_{n \geq 1}$ be a sequence of independent and identically distributed $d \times d$ random matrices and consider the random walk $G_n = g_n \ldots g_1$. In this talk, we will present some recent progress on large and moderate deviations, Berry-Esseen bound and Edgeworth expansion for the coefficients of $G_n$. We will also give some applications to the study of limit theorems for the first passage time of multivariate perpetuity sequences. Mainly based on joint work with I. Grama, Q. Liu and S. Mentemeier.  

We consider a spatial kinetic-type evolution equation, which is used as a model for particle interactions in an ideal gas. Under some regularity assumptions, we investigate its time-dependent and stationary solutions. As key element in our research, we use a branching random walk, which describes changes in particle's velocity after collisions. The talk is based on the work-in-progress with Sebastian Mentemeier.