神马福利影片

Applied Probability Seminar (2025-26)

The 2025鈥�26 Warwick Statistics Applied Probability Seminar will be held on Fridays 11am鈥�12pm in MB0.08. We will join afterwards for coffee/lunch in the Statistics Common Room at 12pm. Everyone is welcome. Please email if you would like to speak or invite a speaker. For visiting speakers: see here for a view of the room.

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Term 3

(Week 1) Fri, May 1 @ MB0.08, 11am鈥�12pm
Speaker: (Warwick, Stats)
Title: The on/off Brownian snake
Abstract: We define what we call an on/off Brownian snake. We use this to construct on/off super Brownian motion recently introduced to the literature by Blath and Jacobi and which is a measure-valued branching process with a dormant state and an active state. Our construction mirrors the construction of super Brownian motion from the Brownian snake by Le Gall. We use the on/off Brownian snake to obtain results concerning the support, range, and expected total mass of on/off super Brownian motion. We have a preprint on arXiv:
(Week 2) Fri, May 8 @ MB0.08, 11am鈥�12pm
Speaker: Janique KrasnowskaLink opens in a new window (Warwick, Stats)
Title: Coalescence in Multi-type Branching Processes
Abstract: Consider a population evolving as a discrete-time supercritical multi-type Galton-Watson process. Suppose we run the process for T generations, then sample k individuals uniformly from generation T and trace their genealogy backwards in time. In the limiting regime as T gets large, we present a formula for the distribution function of the generation t of the most recent common ancestor in terms of the limiting distribution of the normalised population size. We also obtain effective bounds for the decay of this distribution function to 1 in terms of the harmonic moments of the population size at generation t. In order to better understand the behaviour of these harmonic moments, we use a multi-type generalisation of the Harris-Sevastyanov transformation to express harmonic moments at generation t in terms of moments of the transformed process at the first generation.
(Week 3) Fri, May 15 @ MB0.08, 11am鈥�12pm
Speaker: Adam JohansenLink opens in a new window (Warwick, Stats)
Title: Theoretical Properties of Divide-and-conquer Sequential Monte Carlo
Abstract: The divide-and-conquer sequential Monte Carlo algorithm was proposed as a way to facilitate efficient and distributed Monte Carlo computation for large models in 2016. It's theoretical analysis requires the control of errors arising when taking products of empirical measures and propagating them through a recursive algorithm; this talk will demonstrate that such methods inherit many of the desirable properties of standard sequential Monte Carlo methods. Joint work with Juan Kuntz and Francesca Crucinio, which is reported in .
(Week 4) Fri, May 22 @ MB0.08, 11am鈥�12pm
Speaker: (Amsterdam, Maths)
Title: Approximating path-valued functions with signatures in infinite dimensions
Abstract: It is a classical result that functions on a compact set of the real line can be approximated by polynomials. This result can be generalized to functions of paths, in which case they can be approximated by a linear combination of the signature, an object consisting of iterated integrals. This is particularly useful for stochastic differential equations, which can be interpreted as differential equations along random paths. In this talk, I will explain these ideas, and talk about my own research, which is a generalization of these concepts to the infinite dimensional case.
(Week 5) Fri, May 29 @ MB0.08, 11am鈥�12pm
Speaker: Kevin HuangLink opens in a new window (Warwick, Stats)
Title: Large sums of products of random matrices, random energy model and deep and wide neural networks
Abstract: In the study of scaling laws for neural networks, a recurring mathematical object is the infinite product of large random matrices, used to approximate fully connected neural networks with width $n$ and depth $N$ simultaneously growing to infinity. As the width and depth limits do not commute, the joint limit of $n,N \rightarrow \infty$ unveils many intriguing spectral behaviours that differs from those predicted by ergodic theory (for fixed width and growing depth) or by free probability (for fixed depth and growing width). Motivated by the use of residual connections and ensembling techniques in practical machine learning, we seek to understand the effect of incorporating the addition operation in such models. We study the simple model of large empirical averages of large products of large random matrices, in the joint limit where the number of terms to average, $m$ , as well as $n$ and $N$ grow simultaneously. In this triple scaling regime, we discover that the top singular values of this model can be understood through the lens of a random energy model. This allows us to identify an inverse temperature parameter, fully described by $m$ , $n$ and $N$ , that quantifies the scale of the singular values and identifies a phase transition akin to known results in the Gaussian random energy model. This is a joint work with Boris Hanin (Princeton).
(Week 6) Fri, Jun 5 @ MB0.08, 11am鈥�12pm
Speaker: (Birmingham, Maths)
Title: Percolation on the permutahedron and other high-dimensional symmetric graphs
Abstract: In their ground-breaking paper on random graphs, Erd艖s and R茅nyi showed that the component structure of the binomial random graph changes dramatically near to the critical point $p=1/n$ . A similar phase transition has been observed in a number of related random graph models, where under the right scaling their broad-scale structure near to the percolation threshold quantitatively resembles that of $G(n,p)$ , a phenomenon which is referred to as universality. One particular case which has been well-studied is that of the percolated hypercube. The hypercube arises naturally in many combinatorial contexts, in part due to its many equivalent representations (geometric/algebraic/order theoretic). In this talk I will discuss some recent results about this universality in other classes of graphs which come with some underlying high-dimensional geometric or algebraic structure, with a particular focus on the case of the permutahedron, which like the hypercube has many equivalent representations - as the convex hull of the set of permutation vectors, as the dual zonotope dual of the braid arrangement, as the Cayley graph of the symmetric graph generated by adjacent transpositions (notably, a Coxeter system) or as the cover graph of the weak Bruhat lattice. This is joint work with Mauricio Collares and Joseph Doolittle.
(Week 7) Fri, Jun 12 @ MB0.08, 11am鈥�12pm
Speaker: Rubio KouLink opens in a new window (Warwick, Maths)
Title:
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(Week 8) Fri, Jun 19 @ MB0.08, 11am鈥�12pm
Speaker: (Oxford, Stats)
Title:
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(Week 9) Fri, Jun 26 @ MB0.08, 11am鈥�12pm
Speaker: (Imperial, Stats)
Title:
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(Week 10) Fri, Jul 3 @ MB0.08, 11am鈥�12pm
Speaker: Jiayao ShaoLink opens in a new window (Warwick, Stats)
Title:
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Term 2

(Week 1) Fri, Jan 16 @ MB0.08, 11am鈥�12pm
Speaker: (Warwick, Stats)
Title: Brownian motion on spaces of polygons
Abstract: We introduce and study Brownian motion on spaces of discrete regular curves in Euclidean space, such as polygons in the plane. Previously, it has been established when these spaces of discrete regular curves are geodesically complete. I will present joint work with Emmanuel Hartman which relies on a general result by Grigor'yan and shows that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, that is, the associated Brownian motion exists for all times. This provides a rigorous footing for performing data statistics, such as data inference and data imputation, on these spaces. We include simulations for sample paths of Brownian motion on spaces of discrete regular curves.
(Week 2) Fri, Jan 23 @ MB0.08, 11am鈥�12pm
Speaker: Charilaos EfthymiouLink opens in a new window(Warwick, CS)
Title: On Sampling Two Spin Models Using the Local Connective Constant
Abstract: This work establishes novel optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models. These bounds are expressed in terms of the local connective constant of the underlying graph G. This is a notion of effective degree for G, and as such, it allows us to obtain bounds which are inherently less restrictive than those obtained using other graph invariants, e.g., the maximum degree. Our results have some interesting consequences for bounded degree graphs: (A) They include the max-degree bounds as a special case. (B) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, Stefankonic, Yin: PTRF 2017] for general graphs. (C) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on [Hayes: FOCS 2006]. We obtain our results using tools from the theory of high-dimensional expanders and, in particular, the Spectral Independence method [Anari, Liu, Oveis-Gharan: FOCS 2020]. We explore a new direction by utilising the notion of the k-non-backtracking matrix H(G,k) in our analysis with the Spectral Independence. The results with H(G,k) are interesting in their own right. The novelties in the analysis amount to relating the maximum singular value (of a sufficiently large power) of H(G,k) and the spectral radius of the Influence matrix.
(Week 3) Fri, Jan 30 @ MB0.08, 11am鈥�12pm
Speaker: Grega SaksidaLink opens in a new window(Warwick, Maths)
Title: Correlation functions of the quantum XY model
Abstract: The quantum XY model describes the behaviour of spin particles on a lattice. We study the spin-1/2 system, where at each site the spin measured along any axis can be in two states: "up" or "down". The model is quantum, meaning the spin at each site is a complex linear combination of the two possible "measured" states. The spins at adjacent sites interact in a way that favours alignment. A natural question is whether this alignment persists across the entire system. This motivates the study of so-called correlation functions. In this talk, I will define the quantum XY model, and present some new results on the correlation functions. I will explain how we derived these results by representing the quantum XY model as an Ising model with some additional interactions. This is an ongoing joint work with Vedran Sohinger and Daniel Ueltschi.
(Week 4) Fri, Feb 6 @ MB0.08, 11am鈥�12pm
Speaker: (Durham, Maths)
Title: Scaling limits of critical FK-decorated maps at q=4
Abstract: The critical Fortuin鈥揔asteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.
(Week 5) Fri, Feb 13 @ MB0.08, 11am鈥�12pm
Speaker: (Oxford, Maths)
Title: Sharp threshold for reconstructing points on the line
Abstract: For a set of n points V on the real line, let each possible edge be present independently with probability p. We call a subset U of V reconstructible if every injection of V into the real line that preserves the distances along the edges also preserves all pairwise distances in U. How large is the size of a largest reconstructible subset? Gir茫o, Illingworth, Michel, Powierski and Scott conjectured that the answer is linear whp when p = (1+c)/n for every c > 0. We show that for every c > 0 whp there exists a reconstructible subset containing (1-o(1)) proportion of the 2-core. Furthermore, we extend that to some functions c:=c(n) tending to 0.
(Week 6) Fri, Feb 20 @ MB0.08, 11am鈥�12pm
Speaker: Seth HardyLink opens in a new window(Warwick, Maths)
Title: Gaussian multiplicative chaos and the Riemann zeta function
Abstract: First introduced by Kahane in the 1980s to model fluid turbulence, Gaussian multiplicative chaos (GMC) is an area of probability theory that involves understanding measures formed by taking the exponential of Gaussian fields. Since then, research on GMC has found numerous applications throughout mathematics, and somewhat surprisingly, it has recently found applications in number theory, specifically in the study of the Riemann zeta function. In this talk I aim to give an introduction to these areas and their fascinating connection.
(Week 7) Fri, Feb 27 @ MB0.08, 11am鈥�12pm
Speaker: (Bath, Maths)
Title: Fluctuations for fully pushed stochastic fronts
Abstract: This talk concerns traveling wave solutions to stochastic reaction diffusion equations, such as those arising in population genetics to model the genetic composition of an evolving spatial population. Different reaction terms are expected to correspond to three varieties of travelling waves, pushed, semi-pushed, and pulled, which exhibit different behaviours under noisy perturbation. I will discuss a work in progress which characterizes the perturbation of the front in the (fully) pushed regime, which includes both bi-stable and mono-stable reactions. Our main result precisely characterizes the limiting process of the position of the front, in a certain scaling regime, as a Brownian motion with drift. This is joint work with Alison Etheridge, Rapha毛l Forien and Sarah Penington.
(Week 8) Fri, Mar 6 @ MB0.08, 11am鈥�12pm
Speaker: Pablo Ramses Alonso MartinLink opens in a new window(Warwick, Stats)
Title: Fractional slow/fast systems: effective dynamics and parameter estimation
Abstract: Effective dynamics for multiscale systems of SDEs, where the variables of interest are perturbed by environmental variables evolving on a much faster time scale, have been studied extensively since Khasminskii鈥檚 pioneering work in the 1960s. Closely related is the problem of parameter estimation for the resulting effective dynamics, which is crucial for applications. Over the past few decades it has been shown that, for a broad class of models of this sort, subsampling can restore consistency for standard estimators. More recently, there has been a surge of results on effective dynamics for slow/fast stochastic systems that depart from the classical SDE setting by incorporating fractional/rough behaviour. Parameter estimation for these systems, however, remains largely unexplored. In this talk I will present results in this direction by considering a class of random ODEs for which a rough homogenisation (noise-creation) result is known. These models are of particular interest as the effective limit has different qualitative behaviour depending on the parameters of the system. We thus consider the problem of inferring the effective diffusivity and self-similarity using discrete-time observations. By analysing the quadratic variation of additive functionals of a Gaussian processes we show consistency and a non-Central Limit Theorem under appropriate subsampling for a diffusivity estimator, retrieving known results for quadratic variations of self-similar processes. As a consequence, we obtain a consistent estimator for the self-similarity index of the limiting process. Time permitting, I will also mention ongoing work on an averaging principle for a different class of stochastic dynamics that also departs from the Semimartingale and Markovian settings, namely stochastic Volterra equations.
(Week 9) Fri, Mar 13 @ MB0.08, 11am鈥�12pm
Speaker: Wilfrid KendallLink opens in a new window(Warwick, Stats)
Title: Reflections on the Crofton formula
Abstract: The Crofton formula (the length of an arc can be estimated by the number of hits by a random line) is extremely important in modern-day microscopy. Exemplifying Stigler's law of eponymy (itself actually due to Robert Merton), the formula is more properly described as the Cauchy-Crofton formula (announced by Cauchy inComptes Rendusin 1837), and originates from the famous thought experiment on Buffon's baguettes. Together with the associated Poincar茅 kinematic formula, Crofton's formula can be proved by a simple intuitive visual argument in the case of polygonal curves. But the Crofton and Poincar茅 results actually hold for the far more general case of curves of finite total variation, in which context they require rather more technical proofs. I will report on progress I have made in deriving intuitive visual arguments for the general case.
(Week 10) Fri, Mar 20 @ MB0.08, 11am鈥�12pm
Speaker: (UCL, Maths)
Title: Resilience for colour-biased Hamiltonicity in random graphs
Abstract: We begin with the classical Dirac's theorem and the Hamiltonicity threshold result for random graphs, before turning to several common generalizations. In particular, we discuss 'resilience' results concerning the Hamiltonicity of random graphs. We then consider a 'discrepancy' version of Dirac's theorem established by Balogh鈥揅saba鈥揓ing鈥揚luh谩r, by Freschi鈥揌yde鈥揕ada鈥揟reglown, and by Gishboliner鈥揔rivelevich鈥揗ichaeli. We next discuss a further generalization of these results in which we asymptotically determine the maximum value $f_{r,\alpha}(n)$ for every $\alpha \in [\frac{1}{2}, 1]$ such that every $r$ -edge-colouring of every $n$ -vertex graph with minimum degree at least $\alpha n$ contains a Hamilton cycle where one colour appears at least $f_{r,\alpha}(n)$ times. Finally, we present a 'random' analogue of this result, which can be viewed as a generalization of the aforementioned 'resilience' results. This talk is based on joint work with Natalie Behague and Jared Leon.

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Term 1

*** Special out-of-term seminar. Note unusual time and place. ***
Tues, Sep 9 @ MS.05, 4鈥�5pm
Speaker: (Auckland, Stats)
Title: Genealogies of samples from stochastic population models
Abstract: Consider some population evolving stochastically in time. Conditional on the population surviving until some large time $T$ , take a sample of individuals from those alive. What does the ancestral tree drawn out by this sample look like? Some special cases were known, e.g. Durrett (1978), Athreya (2012), O鈥機onnell (1995), but we will discuss some more recent advances when sampling from Bienyame-Galton-Watson (BGW) branching processes conditioned to survive. In near-critical or in critical varying environment BGW settings with finite offspring variances, the same universal limiting sample genealogy always appears up to some deterministic time change which only depends on the mean and variance of the offspring distributions. This genealogical tree has the same binary tree topology as the classical Kingman coalescent, but where the coalescent (or split) times are quite different due to stochastic population size effects, with a representation as a mixture of independent identically distributed times. In contrast, in critical infinite variance offspring settings, we find that more complex universal limiting sample genealogies emerge that exhibit multiple-mergers, these being driven by rare but massive birth events within the underlying population e.g. `superspreaders鈥� in an epidemic. Our key tool for proofs is a change of measure technique involving $k$ distinguished particles, also known as spines. Some ongoing work, open problems, and potential downstream applications will also be mentioned. This talk is based on collaborative works with Juan Carlos Pardo (CIMAT), Samuel Johnston (Kings College London) in Annals of Probability (2024), with Sandra Palau (UNAM), J. C. Pardo in Annals of Applied Probability (2024), and with Matt Roberts (Bath), S. Johnston in Annals of Applied Probability (2020). I would also like to acknowledge the support of the New Zealand Royal Society Te Ap膩rangi Marsden fund.
*** Special out-of-term seminar. Note unusual time and place. ***
Wed, Sep 24 @ MS.05, 11am鈥�12pm
Speaker: (Bath, Maths)
Title: Graphical models for infinite measures with applications to extremes
Abstract: Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure $\Lambda$ on the punctured Euclidean space $\mathbb R^d\setminus \{0\}$ that explodes at the origin. The importance of such measures stems from their connection to infinitely divisible and max-infinitely divisible distributions, where they appear as L茅vy measures and exponent measures, respectively. We characterize independence and conditional independence for $\Lambda$ in various ways through kernels and factorization of a modified density, including a Hammersley-Clifford type theorem for undirected graphical models. As opposed to the classical conditional independence, our notion is intimately connected to the support of the measure $\Lambda$ . Our general theory unifies and extends recent approaches to graphical modeling in the fields of extreme value analysis and L茅vy processes. Our results for the corresponding undirected and directed graphical models lay the foundation for new statistical methodology in these areas. Joint work with Sebastian Engelke, Jevgenijs Ivanovs; accepted in Annals of Applied Probability. Preprint: .
(Week 1) Fri, Oct 10 @ MB0.08, 11am鈥�12pm
Speaker: Oleg ZaboronskiLink opens in a new window (Warwick, Maths)
Title: On the structure of coalescing Ito's diffusions
Abstract: We consider a system of coalescing Ito's diffusion on the real line starting in the maximal entrance law. The corresponding stochastic process generalises both the celebrated Arratia flow as well as the Arratia flow with drift. We show that the one-dimensional distributions are a Pfaffian point process and characterise its kernel as the unique solution to a two-dimensional parabolic equation in half-plane. We apply the Pfaffian structure to the study of the invariant measures for the process. In particular we find that the invariant measure for the unit-variance diffusions with the linear drift $V(x)=-x$ which pushes particles towards the origin is given by the the Pfaffian point process corresponding to the law of real eigenvalues in the real Ginibre ensemble of random matrices. We also study the space of invariant measure for the unit diffusion and the family of algebraic drifts $V(x)=-\mbox{sign}(x) |x|^\alpha$ , $\alpha \in [0,1)$ . We find that at $\alpha=1/3$ the process undergoes a phase transition from the unique (one-particle) steady state to a multi-state phase. Work in progress in collaboration with Roger Tribe, Mykola Vovchansky and Andrey Dorogovtsev.
(Week 2) Fri, Oct 17 @ MB0.08, 11am鈥�12pm
Speaker: Isabella Gon莽alves de AlvarengaLink opens in a new window (Warwick, Stats)
Title: The rightmost particle of the contact process on random dynamical environments
Abstract: The contact process with inherited sterility provides a probabilistic framework for studying population control strategies inspired by the Sterile Insect Technique. Unlike full sterilization, where treated males lose competitiveness, the inherited sterility method introduces only partial sterility that is passed on to descendants, allowing the suppressive effect to propagate across generations. To analyse this model, and to compare it with related dynamics, we also introduce the Spont process, another example of a contact process in a random dynamical environment. We will define the dynamics of both processes on the one-dimensional integer lattice. In both cases, our main result is a central limit theorem for the position of the rightmost occupied site. The two models pose distinct challenges 鈥� the Spont process lacks self-duality, while the inherited sterility model is non-attractive. Our approach combines a renewal-time construction with a careful analysis of active infection paths, leading to an embedded i.i.d. structure for the increments of the position of the rightmost occupied site.
(Week 3) Fri, Oct 24 @ MB0.08, 11am鈥�12pm
Speaker: (Oxford, Maths)
Title: Finding the Origin of a Random Walk
Abstract: Suppose you are given the trace of a symmetric random walk on the integer lattice up to the $n$ -th step, but without any information about the lattice鈥檚 labels. Can you identify the starting point of the walk with probability bounded away from zero? Interestingly, the answer depends on the dimension of the lattice. What if you are given only the set of visited vertices? What about the entire (infinite) trace? I will discuss these phenomena and related problems. Based on joint work with Ritesh Goenka and Peter Keevash.
(Week 4) Fri, Oct 31 @ MB0.08, 11am鈥�12pm
Speaker: (Oxford, Stats)
Title: From $1/\sqrt{n}$ to $1/n$ : Accelerating SDE Simulation with Cubature Formulae
Abstract: Monte Carlo sampling is the standard approach for estimating properties of solutions to stochastic differential equations (SDEs), but its error decays only as $1/\sqrt{n}$ , requiring huge sample sizes. Lyons and Victoir (2004) proposed replacing independently sampled Brownian driving paths with "cubature formulae", deterministic weighted sets of paths that match Brownian "signature moments" up to some degree $D$ . They prove that cubature formulae exist for arbitrary $D$ , but explicit constructions are difficult and have only reached $D=7$ , too small for practical use. We present an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree, reproducing $D=7$ in seconds and reaching $D=17$ within hours on modest hardware. In simulations across multiple SDEs, our cubature formulae achieve an error roughly of order $1/n$ , orders of magnitude smaller than Monte Carlo with the same number of paths. Based on joint work with Thomas Coxon and James Foster.
(Week 5) Fri, Nov 7 @ MB0.08, 11am鈥�12pm
Speaker: John FernleyLink opens in a new window (Warwick, Stats)
Title: The grass-bushes-trees process on a scale-free network
Abstract: The grass-bushes-trees process is a two-type contact process in which one type (the trees),of infection parameter $lambda_1$ , can invade the other type (the bushes) of infection parameter $lambda_2$ . We look to show which graph parameters lead to the possibility of coexistence versus the necessity of competitive displacement, i.e. joint metastability or fast extinction of the bushes. Work in progress with Daniel Valesin.
(Week 6) Fri, Nov 14 @ MB0.08, 11am鈥�12pm
Speaker: Andreas KyprianouLink opens in a new window (Warwick, Stats)
Title: The Brownian marble
Abstract: Fundamentally motivated by the two opposing phenomena of fragmentation and coalescence, we introduce a new stochastic object which is both a process and a geometry. The Brownian marble is built from coalescing Brownian motions on the real line, with further coalescing Brownian motions introduced through time in the gaps between yet to coalesce Brownian paths. The instantaneous rate at which we introduce more Brownian paths is given by $\lambda/g^2$ where $g$ is the gap between two adjacent existing Brownian paths. We show that the process 鈥渃omes down from infinity鈥� when $0<\lambda<6$ and the resulting space-time graph of the process is a strict subset of the Brownian Web on ${\mathbb R} \times [0,\infty)$ . When $\lambda\geq 6$ , the resulting process 鈥渄oes not come down from infinity鈥� and the resulting range of the process agrees with the Brownian Web.
(Week 7) Fri, Nov 21 @ MB0.08, 11am鈥�12pm
Speaker: Gareth RobertsLink opens in a new window (Warwick, Stats)
Title: Ballistic and diffusive lifted MCMC, with application to parallel tempering
Abstract: In this talk I will review the popular 鈥渓ifting鈥� mechanism for producing non-reversible Markov chain Monte Carlo such as non-reversible Metropolis-Hastings and piecewise-deterministic Markov processes. These methods aim to have better mixing by providing momentum to break down random walk behaviour of algorithms. The presentation will investigate how these behave in a collection of stylised high-dimensional examples showing that the non-reversibility can often be washed out by the problem complexity so that the algorithm behaves asymptotically in a reversible way. On the other hand lifted algorithms still retain a small efficiency advantage over their reversible counterparts. Furthermore, we will show that some carefully constructed higher-order lifted Metropolis-Hastings algorithms can retain some aspects of ballistic behaviour, even in the high-dimensional limit setting.
(Week 8) Fri, Nov 28 @ MB0.08, 11am鈥�12pm
Speaker: Oleg PikhurkoLink opens in a new window (Warwick, Maths)
Title: Randomness for ball covering of large-dimensional Euclidean spaces
Abstract: We will discuss the power and limitations of using randomness for proving the existence of coverings of ${\mathbb R}^n$ , for large $n$ , by Euclidean unit balls with small density (which is, informally speaking, the number of balls that contain a `typical' point of ${\mathbb R}^n$ ). This talk will be based on joint work with Boris Bukh, Jun Gao, Xizhi Liu and Shumin Sun ( and ).
(Week 9) Fri, Dec 5 @ MB0.08, 11am鈥�12pm
Speaker: Ian MelbourneLink opens in a new window (Warwick, Maths)
Title: Convergence to Levy processes for deterministic dynamical systems
Abstract: I will survey results over the last 10 years on convergence to Levy processes for deterministic dynamical systems. Convergence to a Levy process often holds when the central limit theorem fails. The limiting process is superdiffusive (growing like (time)^a with a>1/2) and sample paths have dense sets of discontinuities. Classical treatments of convergence to Levy processes use Skorokhod topologies from 1956. In the 1990s, Whitt recognised that convergence may fail in such topologies, and that important information may be lost even when convergence holds. Accordingly, Whitt introduced "decorated" Skorokhod-type topologies. However, there was a lack of examples to illustrate how best to proceed. It turns out that dynamical systems provide a wealth of examples where decorated topologies are needed. Moreover, their analysis leads to the correct (we claim!) definition of decorated Skorokhod topology. The precise definitions are technical. Instead I'll provide examples and pictures to illustrate the theory. This is joint work with Chevyrev & Korepanov and with Freitas, Freitas & Todd.
(Week 10) Fri, Dec 12 @ MB0.08, 11am鈥�12pm
Speaker: Vedran SohingerLink opens in a new window (Warwick, Maths)
Title: Gibbs measures as local equilibrium Kubo-Martin-Schwinger states for focusing nonlinear Schr枚dinger equations
Abstract: Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure global well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the classical Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari (University of Besan莽on, Bourgogne-Franche-Comt茅). In the second part of the talk, we study Gibbs measures for focusing nonlinear Schr枚dinger equations, where the Hamiltonian is no longer positive definite. The Gibbs measures now have to be localised by a truncation in the mass or renormalised mass, depending on the dimension. We show that local Gibbs measures correspond to suitably localised KMS states. This is joint work with Andrew Rout (Politecnico di Milano) and Zied Ammari (University of Besan莽on, Bourgogne-Franche-Comt茅).

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