Abstracts
Tibor Antal
University of Edinburgh
Title: Sequential mutations in exponentially growing populations
Abstract: Stochastic models of sequential mutation acquisition are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells are there with n alterations, and how long will it take for these cells to appear in an exponentially growing populations. In this talk, within a multitype branching process framework, we consider a general mutational path where mutations may be advantageous, neutral or deleterious [1]. In the biologically relevant limiting regimes of large times and small mutation rates, we derive probability distributions for the number, and arrival time, of cells with n mutations. Surprisingly, the two quantities respectively follow Mittag-Leffler and logistic distributions regardless of n or the mutations’ selective effects. Extensions to mutational processes where cell types can change into several neighbouring types on a connectivity graph. and applications to mutation accumulation in healthy tissue on the way to cancer [3] will also be discussed.
[1] Sequential mutations in exponentially growing populations MD Nicholson, D Cheek, T Antal PLOS Computational Biology 19 (7), e1011289 (2023)
[3] Accumulation of Oncogenic Mutations During Progression from Healthy Tissue to Cancer R Zhang, I Bozic - bioRxiv 2024.02.18.580841, (2024) - biorxiv.org
Márton Balázs
University of Bristol
Title: Road layout in the KPZ class Abstract: In this talk I will be after a model for road layouts. Imagine a Poisson process on the plane for start points of cars. Each car picks an independent random direction and goes straight that way for some distance. I will start with showing that the origin (my house, that is) will see a lot of car traffic within an arbitrary small distance.
Which is not what we find in the real world out there, why? The answer is of course coalescence of paths in the random environment provided by hills and other geographic or societal obstacles. This points us towards first passage percolation (FPP), expected to be a member of the KPZ universality class. Due to lack of results for FPP, instead we built our model in exponential last passage percolation (LPP), known to be in the KPZ class. I will introduce LPP, then explain how to construct our road layout model in LPP, and what phenomena we can prove about roads and cars in this model using results from the LPP literature. (joint with Riddhipratim Basu, Sudeshna Bhattacharjee, Karambir Das, David Harper)
Julien Berestycki
University of Oxford
Title: Branching selection particle systems and the selection principle.
Abstract: The \(N\)-branching Brownian motion with selection (\(N\)-BBM) is a particle system consisting of \(N\) independent particles that diffuse as Brownian motions in \(\mathbb{R}\), branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. It is a very simple model for the evolution of a population under selection that has generated some fascinating research since its introduction by Brunet and Derrida in the early 2000’s.
If one recentre the positions by the position of the left most particle, this system has a stationary distribution. I will show that, as \(N\)\ra\(\infty\) the stationary empirical measure of the \(N\)-particle system converges to the minimal travelling wave of an associated free boundary PDE. This resolves an open question going back at least to works of e.g. Maillard in 2012. It follows a recent related result by Oliver Tough (with whom this is joint work) establishing a similar selection principle for the so-called Fleming-Viot particle system.
Leonid Bogachev
University of Leeds
Title: Large Components in Random Decomposable Structures and the Bose-Einstein Condensation
Abstract: Random decomposable structures (such as integer partitions or permutations) may manifest the emergence of a “giant” component due to the asymptotic loss of mass. Connections of such models with statistical physics are well known. Work by A.M.,Vershik stimulated interest in the limit shape of large combinatorial structures; in particular, he proposed an interpretation of limit shape in terms of the Bose–Einstein condensation of ideal quantum gas. In my talk, I will present two results in this direction: one related to a unified derivation of the limit shape of random partitions, and the other about the asymptotic statistics of cycles in the model of “surrogate-spatial” permutations, proposed as an analytically tractable version of the model of spatial random permutations.
Jullianna Bor
Imperial College, London
Title: Response time in a pair of processor sharing queues with Join-the-Shortest-Queue scheduling
Abstract: College London, which focuses on the application of stochastic techniques to performance analysis and engineering problems in computing services. I will then present a recent result deriving a numerical solution for the response time distribution in two parallel processor sharing (PS) queues with Join-the-Shortest-Queue (JSQ) scheduling for the first time.
JSQ is attracting renewed attention within the computer systems community for load-balancing across networks and cloud servers. Our work examines it in the more challenging case where individual queues are served under the PS queueing discipline. Using the generating function method, two partial differential equations are obtained corresponding to conditional response times, which are solved using a novel numerical technique. Numerical results for the density and the first four moments show high accuracy compared to simulation.
Ofer Busani
University of Edinburgh
Title: On the number of infinite geodesics at exceptional directions in the KPZ class
Abstract: The KPZ class consists of many models of random growth interface. In many cases, the dynamics of these models can be studied via variational forms that give rise to metric-like spaces, which in turn, can be studied through geodesics. The study of infinite geodesics in the KPZ class has been studied intensively in the past 30 years. One central question is the following:
Given a direction, how many infinite geodesics that are asymptotically going in that direction are there?
In this talk I shall discuss what we know about infinite in the KPZ class and some recent developments regarding the question above.
Based on several works with Marton Balazs, Timo Seppalainen and Evan Sorensen.
Florin Ciucu
University of Warwick
Title: On Some Ultra-Sharp Bounds in Queueing Systems
Abstract: The talk introduces two new classes of ultra-sharp bounds in queueing systems. The first is specific to the G/G/1 queue, with possibly correlated inter-arrivals; the main result is an exact expression for the distribution of the delay in terms of a series, whose terms are subject to elementary integration. Remarkably, the first few terms are sufficient to render ultra-sharp bounds improving upon state-of-the-art bounds by orders of magnitude. The second new class of bounds is specific to a tandem queueing network with general (i.e., not necessarily Poisson) arrivals. Besides showing for the first time that the end-to-end delay distribution is subject to a polynomial-exponential structure, explicit bounds computed in some special cases are shown to improve upon state-of-the-art results by many orders of magnitude.
Konstantinos Dareiotis
University of Leeds
Title: Regularisation by noise in rough differential equations
Abstract: We show strong well-posedness of rough differential equations with distributional drift driven by the Gaussian rough path lift of fractional Brownian motion with Hurst parameter \(H\in(1/3,1/2)\). The noise coefficient is uniformly elliptic and sufficiently regular while the drift is a distribution in the Holder-Besov space \(\mathcal{C}^\alpha\) with \(\alpha>1-1/(2H)\). The latter condition matches the one of the additive noise setting from [Catellier-Gubinelli, 2016] thereby providing a complete multiplicative analogue. This is joint work (in progress) with M. Gerencsér, K. Lê, and C. Ling.
Denis Denisov
University of Manchester
Title: Ordered Random Walks
Abstract: We study a d-dimensional random walk with independent and identically distributed increments in a Weyl chamber. We discuss the 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 the number of walks grows with time.
Ayalvadi Ganesh
University of Bristol
Title: Large deviations for Cox processes and Cox/G/infinity queues, with a biological application
Abstract: We show that a sequence of Cox processes on a Polish space E satisfy a large deviation principle (LDP), provided their directing measures do so on the space of finite measures on E equipped with the weak topology. Next, we consider a sequence of infinite server queue with general iid service times, where the arrivals constitute Cox processes with translation invariant directing measures assumed to satisfy an LDP. We show that the corresponding sequence of queue occupancy measures also satisfy an LDP. These questions were motivated by the problem of describing fluctuations of molecule numbers in biochemical reaction networks within cells.
Joint work with Justin Dean and Edward Crane.
Robert Gaunt
University of Manchester
Title: Normal approximation for the posterior in exponential families
Abstract: The Bernstein-von Mises (BvM) Theorem is a cornerstone in Bayesian statistics. Loosely put, this theorem reconciles Bayesian and frequentist large sample theory by guaranteeing that, under regularity conditions, suitable scalings of posterior distributions are asymptotically normal. In particular, this implies that the contribution of the prior vanishes in the asymptotic posterior.
In this talk, we demonstrate how the probabilistic technique Stein’s method can be used to derive explicit optimal order total variation and Wasserstein distance bounds to quantify this distributional approximation for posterior distributions in exponential family models. We apply our general bounds to some classical conjugate prior models and observe that the resulting bounds have an explicit dependence on the prior distribution and on sufficient statistics of the data from the sample, and thus provide insight into how these factors may affect the quality of the normal approximation.
This is joint work with Adrian Fischer, Gesine Reinert and Yvik Swan
István Gyöngy
Universtiy of Edinburgh
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Jonathan Jordan
University of Sheffield
Title: Competing types in preferential attachment graphs with community structure
Abstract: We extend the two-type preferential attachment model of Antunović, Mossel and Rácz to networks with community structure. We show that different types of limiting behaviour can be found depending on the choice of community structure and type assignment rule.
In particular, we show that, for essentially all type assignment rules where more than one limit has positive probability in the unstructured model, communities may simultaneously converge to different limits if the community connections are sufficiently weak. If only one limit is possible in the unstructured model, this behaviour still occurs for some choices of type assignment rule and community structure. However, we give natural conditions on the assignment rule and, for two communities, on the structure, either of which will imply convergence to this limit, and each of which is essentially best possible.
Although in the unstructured two-type model convergence almost surely occurs, we give an example with community structure which almost surely does not converge.
This is joint work with John Haslegrave and Mark Yarrow.
Wilfrid Kendall
University of Warwick
Title: Wandering around a fibrous network when all the paths look very much alike
Abstract: I will give an overview of work I have been interested in for the last decade or so, concerned with generating models based on Poisson processes which supply insight into route-finding and traffic flows. Initial constructions (Aldous 2014, Kahn 2016, Kendall 2017) showed existence of such models satisfying the SIRSN axioms formulated by Aldous (2014). But these constructions all involve infinitely long linear paths, and in particular do not easily admit local influences. Can one do better? I shall discuss recent work showing that SIRSN can indeed arise under far less stringent conditions, based on line segments or even suitably stiff fibres. References: Aldous, D J. “Scale-Invariant Random Spatial Networks.” Electronic Journal of Probability 19, no. 15 (April 2014): 1-41. https://doi.org/10.1214/EJP.v19-2920. Kahn, J. “Improper Poisson Line Process as SIRSN in Any Dimension.” Annals of Probability 44, no. 4 (July 2016): 2694-2725. https://doi.org/10.1214/15-AOP1032. WSK “From Random Lines to Metric Spaces.” Annals of Probability 45, no. 1 (2017): 469-517. https://doi.org/10.1214/14-AOP935.
Dmitry Korshunov
University of Lancaster
Title: Large deviations for asymptotically space homogeneous Markov chains in two dimensions
Abstract: We discuss Markov chains in positive lattice quadrant whose transition probabilities stabilise at infinity. Assuming positive recurrence of the chain we study large deviations of the invariant probabilities under Cramer type conditions on jumps.
It is a joint work with D Denisov and V Wachtel
Khoa Lê
University of Leeds
Title: Davie’s uniqueness for stochastic differential equations under Krylov–Röckner condition
Abstract: We show that any stochastic differential equation (SDE) driven by Brownian motion with drift satisfying the Krylov–Röckner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion. Additionally, we show that such SDE is strongly complete, i.e. for almost every trajectory of the Brownian motion, the family of solutions with different initial data forms a continuous semiflow for all nonnegative times. This is a joint work with Lukas Anzeletti and Chengcheng Ling.
Jiawei Li
University of Edinburgh
Title: Twin Brownian particle method for the study of Oberbeck-Boussinesq fluid flows
Abstract: We establish stochastic functional integral representations for solutions of Oberbeck-Boussinesq equations in the form of McKean-Vlasov-type mean field equations, which can be used to design numerical schemes for calculating solutions and implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for the solution of some linear parabolic equation in terms of the Green function and the pinned measure of the associated diffusion. We demonstrate via numerical experiments the efficiency of the numerical schemes, which are capable of revealing numerically the details of Oberbeck-Boussinesq flows within their thin boundary layer, including Benard’s convection feature.
Gar Goei Loke
University of Durham
Title: An optimizer’s perspective of queueing theory: Model primitives for solving capacity, routing, priority and inflow control decisions in queueing networks in polynomial time
Abstract: We propose a novel framework for solving optimization problems posed in the setting of queueing networks that involve optimizing capacity, routing across nodes, inflow control and time-based priority policies. Our framework involves introducing a completely new set of modelling primitives for queueing dynamics in the discrete transient finite-time horizon setting. In particular, our framework leverages on the introduction of a new index that tracks the amount of time a job has spent in each node. This results in a ‘lifting’ of the dynamics where queue metrics, such as queue lengths and (average) waiting times, are expressed linearly in the state variables. It can be solved as a sequence of convex optimization problems, and thus is polynomial time complex, despite solving for multi-period adaptive policies under general distributions of arrivals, service completions and time-based abandonment. The framework has been applied to a series of applications involving scheduling, dynamic capacity allocation, dynamic demand-supply matching and inventory control, and has been shown to outperform approximate dynamic programming, fluid approximations, sample average approximations and threshold policies in terms of out-of-sample performance in various numerical experiments.
Mateusz Majka
Heriot-Watt University
Title: Sampling, optimization, SDEs and gradient flows
Abstract: We discuss connections between the problem of approximate sampling from a given probability measure and the problem of minimizing functions defined on the space of probability measures motivated by machine learning applications. For both these problems, one can construct continuous-time stochastic processes that converge to the target measure as time goes to infinity. Among many ways of studying convergence rates for such processes, we focus on the approach via functional inequalities. In particular, we discuss the Polyak-Lojasiewicz inequality on the space of measures and its relation to the classical log-Sobolev inequality, and we explain why it is a natural condition for obtaining exponential convergence of Fisher-Rao gradient flows. We also mention applications of such flows to solving min-max problems on the space of probability measures, motivated by the problem of training Generative Adversarial Networks. Based on joint papers with Razvan-Andrei Lascu (Heriot-Watt), Linshan Liu (Heriot-Watt) and Lukasz Szpruch (University of Edinburgh).
Aleks Mijatovic
University of Warwick
Title: Subexponential lower bounds for \(f\)-ergodic Markov processes
Abstract: In this talk I will describe a criterion for establishing lower bounds on the rate of convergence in \(f\)-variation of a continuous-time ergodic Markov process to its invariant measure. The criterion consists of novel super- and submartingale conditions for certain functionals of the Markov process. It provides a general approach for proving lower bounds on the tails of the invariant measure and the rate of convergence in \(f\)-variation of a Markov process, analogous to the widely used Lyapunov drift conditions for upper bounds. Our key technical innovation, which will be discussed in the talk, produces lower bounds on the tails of the heights and durations of the excursions from bounded sets of a continuous-time Markov process using path-wise arguments.
I will present applications of our theory to elliptic diffusions and Levy-driven stochastic differential equations with known polynomial/stretched exponential upper bounds on their rates of convergence. Our lower bounds match asymptotically the known upper bounds for these classes of models, thus establishing their rate of convergence to stationarity. The generality of our approach suggests that, analogous to the Lyapunov drift conditions for upper bounds, our methods can be expected to find applications in many other settings. This is joint work with Miha Brešar at Warwick.
Ilya Molchanov
University of Bern
Title: Isoperimetry and Extremal Processes
Abstract: The talk deals with asymptotic results for distributions of homogeneous functionals of min-stable or max-stable processes and for their conditional distributions given a homogeneous functional. As a simple application, one obtains results for the maximum of the process given that its infimum is large.
This is a joint work with Daniel Hug and Kirstin Strokorb.
John Moriarty
Queen Mary University of London
Title: Optimal stopping with nonlinear expectation: geometric and algorithmic solutions
Abstract: We use the geometry of functions associated with martingales under nonlinear expectations to solve risk-sensitive Markovian optimal stopping problems. Generalising the linear case due to Dynkin and Yushkievich (1969), the value function is the majorant or pointwise infimum of those functions which dominate the gain function. An emphasis is placed on the geometry of the majorant and pathwise arguments, rather than exploiting convexity, positive homogeneity or related analytical properties. An algorithm is provided to construct the value function at the computational cost of a two-dimensional search. The talk is based on the preprint https://arxiv.org/abs/2306.17623 (with Tomasz Kosmala).
Arpan Mukhopadhyay
University of Warwick
Title: On Optimal Server Allocation for Parallelisable Jobs
Abstract: A large proportion of jobs submitted to modern computing clusters and data centers are parallelisable and capable of running on a flexible number of computing cores or servers. Examples of such jobs include training of large machine learning models, simulation of climate models etc. Although allocating more servers to such a job results in a higher speed-up in the job’s execution, it reduces the number of servers available to other jobs, which in the worst case, can result in an incoming job not finding any available server to run immediately upon arrival. Hence, a natural question to ask is: how to optimally allocate servers to jobs such that (i) the average execution time across jobs is minimised and (ii) almost all jobs find at least one server immediately upon arrival. To address this question, we consider a loss system where jobs with arbitrary concave speed-up functions arrive and a server allocation scheme is used to allocate one or more servers from the system to each arriving job based on the availability of the servers. If a job does not find any available servers upon entry, then it is blocked and lost. We propose a simple server allocation scheme that achieves the minimum average execution time of accepted jobs while ensuring that the blocking probability of jobs vanishes as the system becomes large. To prove our result, we employ Stein’s method which also yields non-asymptotic bounds on the blocking probability and the mean execution time.
The talk will be based on joint work with Samira Ghanbarian (uWaterloo), Ravi R. Mazumdar (uWaterloo), and Fabrice Guillemin (Orange Labs, France).
Amanda Turner
University of Leeds
Title: Scaling Limits of Random Growth Models
Abstract: Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above, but there are many natural generalizations. In this talk I will give a survey of the main results and conjectures in this area.
Thirupathaiah Vasantam
University of Durham
Title: Performance Analysis of a Quantum Switch: An Erlang Loss Model
Abstract: Quantum internet is an emerging internet technology that will be used to exchange quantum information between quantum processors for communication, cryptographic, and computation purposes. It will have applications that are impossible to achieve using current internet technology. A key requirement for many applications of quantum internet is distribution of entanglements, quantum states that have correlated qubits, to different quantum nodes. In this work, we study a quantum switch, entanglement generation switch (EGS), that distributes entanglements to neighbour nodes. A quantum switch is a core component of quantum networks that make quantum internet. By considering the operating principles of a physical quantum switch, we model it as an erlang loss model with entanglement requests arriving as sessions of a Poisson process with each session having multiple attempts for entanglements. An entanglement request will be served if related nodes and the switch have sufficient resources for creating the requested entanglement, otherwise it will be blocked from service. We derive analytical expressions for blocking probabilities and show that they are insensitive to the service time distributions of entanglement requests. We provide numerical results to support our analysis.
This is a joint work with Scarlett Gauthier (TU Delft) and Gayane Vardoyan (TU Delft & UMass, Amherst)
Vladislav Vysotskiy
University of Sussex
Title: The isoperimetric problem for convex hulls and the large deviations rate functions of random walks
Abstract: We study the asymptotic shape of the most likely trajectories of a planar random walk that result in large deviations of the area of the convex hull of the first \(n\) steps of the walk, as \(n \to \infty\). If the increments of the walk have finite Laplace transform, such a scaled limit trajectory $h$ solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of $h$ is replaced by the large deviations rate functional \(\int_0^1 I(h'(t)) dt\) with \(I\) being the rate function of the increments. Assuming that the distribution of increments is not contained in a half-plane, we show that the optimal trajectories are smooth, convex, and satisfy the Euler–Lagrange equation, which we solve explicitly for every \(I\). Our solution resembles that of the isoperimetric problem in the Minkowski plane found by Busemann (1947).
Andrew Wade
_University of Durham
Title: Iterated-logarithm laws for convex hulls of random walks with drift
Abstract: Consider a random walk whose increments have finite variance, run for a finite number of steps. The convex hull of the trajectory is the (random) minimal convex polytope that encloses all the points visited by the walk. We present some results on laws of the iterated logarithm for intrinsic volumes of the convex hull, in the case where the random walk has a non-zero drift. Analogous results in the case of zero drift (where the scaling is different) were obtained by Khoshnevisan. This talk is based on joint work with Wojciech Cygan, Nikola Sandrić, and Stjepan Šebek.
Neil Walton
University of Durham
Title: Exponential Concentration in Stochastic Approximation
Abstract: We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, progress towards an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tail-bounds contrast asymptotic normality results, which are more frequently associated with stochastic approximation. The methods that we develop rely on a geometric ergodicity proof. This extends a result on Markov chains due to Hajek (1982) to the area of stochastic approximation algorithms. We apply our results to several different Stochastic Approximation algorithms, specifically Projected Stochastic Gradient Descent, Kiefer-Wolfowitz and Stochastic Frank-Wolfe algorithms. When applicable, our results prove faster O(1/t) and linear convergence rates for Projected Stochastic Gradient Descent with a non-vanishing gradient.
MinMin Wang
University of Sussex
Title: Random intersection graphs at criticality
Abstract: We study a random graph model that has a flexible degree distribution and nontrivial clustering. The random graph in question can be constructed from a bipartite graph with a dual sequence of vertex weights. We will focus on the case where the weights are sampled from two power laws. Depending on these two power laws, a range of limit distributions can emerge for the component sizes at the critical threshold for connectivity.
Joint work with Alastair Haig.