We consider one-dimensional transient Markov chains $X=(X_n)$ on a positive half line. Let $H_y(x) = \sum_{n=0}^\infty \mathbf P_y(X_n\le x)$ be the renewal function of $X$. In this talk I will discuss integral and local renewal theorems for $H_y(x),$ as $x\to \infty$.

The talk is based on the following joint works with D. Korshunov and V.Wachtel:
[1] D. Denisov, D. Korshunov, and V. Wachtel. Potential analysis for positive recurrent Markov chains with asymptotically zero drift: Power-type asymptotics, Stoch. Proc. Appl. 123 (2013) 3027–3051.
[2] D. Denisov, D. Korshunov, and V. Wachtel. At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift, arXiv:1612.01592 (2016).

In this talk I describe the long term behaviour of a growth process formed by numbers of particles sequentially deposited at sites of a cycle graph. The model of interest is motivated by cooperative sequential adsorption processes and can also be regarded as a reinforced urn model with graph-based interactions. Our main result consists in showing that with probability one the growth process will eventually localise either at a single site, or at a pair of neighbouring sites. More precisely, our result classifies all the possible behaviours of the model in terms of the set of parameters associated with the site-dependent growth rates. (Joint work with M. Menshikov, V. Shcherbakov and M. Vachkovskaia.)

Dose response models provide valuable information when quantifying the risk to an individual following infection by a pathogen. However, by also considering the time scale of infection, a more detailed set of results can be obtained. In [1], authors propose a within-host stochastic model for Francisella tularensis infection that incorporates key biological mechanisms to define interactions between host phagocytes and extracellular bacteria. From this, the probability of response and mean time until response is computed as a function of the initial infection dose. Despite including within-host reactions, within-phagocyte infection dynamics are not explicitly accounted for, and it is assumed that when each infected phagocyte ruptures, a fixed number of bacteria are released. Here, a multi-scale extension to this model is proposed that links the within-phagocyte and within-host infection dynamics. A Bayesian approach is applied to parametrise the within-phagocyte model using infection data, from which, the probability mass function of the number of bacteria released by a rupturing phagocyte is computed. Linking this distribution into the within-host model, it is shown that, whilst the probability of response is comparable to that obtained in [1], the time until response might have been underestimated.

[1] R.M. Wood, J.R. Egan, and I.M. Hall. A dose and time response Markov model for the in-host dynamics of infection with intracellular bacteria following inhalation: with application to Francisella tularensis, Journal of The Royal Society Interface 11(2014) 20140119.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, we build the shortest fence possible around the garden, but what shape is the garden? We can get some indication by studying the perimeter length and diameter of the convex hull of a planar random walk. We describe the limiting behaviour for these two functionals in both the case of zero and non-zero drift and then study the ratio of these two values using the relationship between random walks and Brownian motion.

We analyse the expectation and variance of the number of Euler tours of a random regular graph. The majority of our work focuses on the directed regular model of $d$-in, $d$-out random graphs, $d \geq 2$. We also show how to obtain the asymptotic distribution and prove a concentration result using the small subgraph conditioning method. Hence we are able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. For this we make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $([(d-1)!]^n)/n$. Therefore most of our work lies in estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph. We don't have equivalent results for the undirected regular model, but all the same we will derive the expected number of tours/arborescences (though not the variance or higher moments). (Joint work with Páidí Creed.)