Batı Şengül
batisengul [at] gmail (dot) com

I am currently a postdoc at King's College London, working with Aleksandar Mijatovic. Prior to this a postdoc at the University of Bath working with Andreas Kyprianou and before that I studied for a PhD at the University of Cambridge with Nathanael Berestycki.

My research interests are in probability theory, specifically I am interested in coalescent/fragmentation processes, mixing times, random graphs, random permutations, self-similar Markov processes and Lévy proccesses.

## Papers

Asymptotic number of caterpillars of regularly varying Lambda-coalescents that come down from infinity
submitted
Exceptional times of the critical dynamical Erdős-Rényi graph
with Matthew Roberts, submitted
Existence of a phase transition of the interchange process on the Hamming graph
with Piotr Miłoś, submitted
Deep factorisation of the stable process II; potentials and applications
with Andreas Kyprianou and Victor Rivero, submitted
Cutoff for conjugacy-invariant random walks on the permutation group
with Nathanaël Berestycki , submitted
Conditioning subordinators embedded in Markov processes
with Andreas Kyprianou and Victor Rivero, SPA (to appear)
Scaling Limit of Coalescent Processes Near Time Zero
Ann. Inst. H. Poincaré (to appear)
Combined First and Second Order Total Variation Inpainting using Split Bregman
[IPOL] [pdf] [code]
with Kostas Papafitsoros and Carola Schoenlieb , Image Processing On Line (2013)

### Lecture notes

Probabilistic approach to coalescent processes
[pdf]

### Projects and essays

Geometric Aspects and Mixing Times of Coalescent and Fragmentation Processes
[pdf]
PHD Thesis (2014)
Estimates for Dynamics of Coupled Flagella
[pdf]
CCA Short Mini-Project (2011)
Random Motions in Spacetime
[pdf]
Part III essay (2010)
Introduction to Levy processes
[pdf]
Third year project (2009)

## Research

### Lévy processes and self-similar Markov processes

A Lévy processes is a Markov process with independent and stationary increments. A Markov process $X=(X_t:t \geq 0)$ is called self-similar of index $\alpha>0$ if for every constant $c>0$, $(cX_{c^{-\alpha}t}:t \geq 0)$ has the same law as $X$. These processes appear in various areas of probability and have applications in financial mathematics. With Andreas Kyprianou and Victor Rivero we looked at conditioning increasing Lévy process to stay below a level in this paper. In particular this has a rather nice form when the increasing Lévy process is self-similar.

Not every self-similar Markov process is a Lévy process. The theory of Lévy process is very well understood and recently there has been a lot of work in trying to transfer this knowledge to self-similar Markov processes. A tool known as Lamperti-Kiu tranformation allows one to write a self-similar Markov process as a time change of a Markov additive process, which can be thought of a mixture of Lévy processes. In order to understand the path properties of self-similar Markov processes, we tried to understand the behaviour of these Markov additive processes. As a first step in that direction, in this paper, myself, Andreas Kyprianou and Victor Rivero look at path properties of self-similar Markov additive process that arrises when the underlying process is a Lévy processes. We were able to show an analogue of the Wiener-Hopf factorisation as well as obtaining some fluctuation identities.

### Coalscent Processes

A coalscent processes is a stochastic particle system in which particles merge together. Following this process along gives rise to a tree, which can be thought of as describing ancestral relationships of individuals alive today. That is, we think of each particle as representing an individual alive today, and when particles merge, we think of them having a common ancestor at that point in time.

There are many models for coalescent processes, the simplest, known as Kingman's coalescent, is when any two pair of particles merge together at rate 1. This model exhibits a big-bang like behaviour: if we start the process with countable infinity many particles, then for any time $t>0$, there are only finitely many particles at time t. This phenomena has also been observed for some of the more general coalescent models as well and is known as coming down from infinity.

In this paper, I look at a subset of coalescent processes that come down from infinity. Specifically, I take scaling limits around time 0 to observe how they behave in the moment they come down from infinity. To do this, I view the coalescent process as a random tree (just like in the picture) and then zoom in around the tips.

### Random walks on the permutation group

Imagine you have a deck of cards that are being shuffled. How many shuffles do you have to do before the deck looks well mixed? This question has been formulated mathematically in the 1980s by Diaconis and Shahshahani (and in an other context at the same time by Aldous) and since then has been studied intensively.

One particular problem studied by Diaconis and Shahshahani is shuffling the deck by swapping two cards chosen uniformly at random at each step and looking at what happens when the size of the deck tends to infinity. We can view this as a random walk on the permutation group, where the walk applies a uniformly chosen transposition at every step. They find that the model exhibits a cut-off: before time $(1/2)n \log n$, the walk is not close to the uniform distribution and after time $(1/2)n \log n$, the walk is arbitrarily close to the uniform distribution. The proof given by Diaconis and Shahshahani use a beautiful link to representation theory (which is like Fourier analysis on groups). Since then, ongoing work has tried to generalise this result. For example, what if the random walk applies a uniformly chosen k-cycle at each step.

In this paper, me and Nathanael Berestycki answer a conjecture about general walks with conjugacy invariant steps varying with n, in particular this covers the case when the random walk applies a uniformly chosen k-cycle at each step with k depending on n. Rather surprisingly, our proof does not use any representation theory, but instead is purely probabilistic. We view the symmetric group a metric space and obtain bounds on its coarse Ricci curvature, which is a recent notion of Ricci curvature on metric spaces introduced here.

### Interchange process

Take an undirected bounded degree graph and place a particle on each vertex. Then at rate 1, select an edge uniformly at random and swap the particles on either end of this edge. We can describe the configuration of the positions of the particles at time t as a permutation $\sigma_t:V\rightarrow V$ on the vertex set of the graph where $\sigma_t(v)$ is the position of the particle at time t which started at vertex v. This is called the interchange process.

The model was introduced in the 1990's by Tóth to answer questions about ferromagnetic models in physics. From a mathematical perspective, it is really challenging to analyse this process. The question of interest is to describe the cycle lengths of this permutation. Schramm gave a detailed description of the cycle lenghs on the complete graph, which included the proof of a phase transition. If $t=cn$, where n is is the size of the complete graph, then as $n\to\infty$, $\sigma_t$ has cycles of length $O(\log n)$ when $c \lt 1/2$ and has cycles of length comparable to n when $c\gt 1/2$.

Since Schramm's work, various alternative proofs of this phase transition on the complete graph has appeared, a similar phase transition has been shown in the case of the infinite regular tree, and a weaker version of this phase transition has been recently shown on the hypercube. Outside of these works, very little is known. In this paper me and Piotr Miłoś develop new tools to show the existence of the phase transition on the Hamming graph. We chose this graph because it is the graph that is the closest to the complete graph where the methods on the complete graph fail to give optimal results.

## Software

A Google Chrome extension which automatically sets a desired quality on YouTube videos.

This is a package written for the Atom.io text editor which implements latex autocompletions.

A Google Chrome extension which displays the rating from IMDB on BBC iPlayer.