Probability section
Invited speakers:
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Contributed talks:
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Programme:
Mon 30th March, 2026
| 3:05pm – 3:30pm | Jessica Jay (Lancaster) |
| “Connections between interacting particle systems and combinatorial objects” (abstract) | |
| 3:35pm – 3:45pm | Gilles Germain (Oxford) |
| “Normal approximation of the intrinsic volumes distribution of a convex body” (abstract) | |
| 3:45pm – 3:55pm | Nero Z. Li (Imperial) |
| “Iterated graph systems: Brownian motion on fractals” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 4:55pm | Tengyao Wang (LSE) |
| “Coverage correlation: detecting singular dependencies between random variables” (abstract) | |
| 4:55pm – 5:05pm | Alberto Bordino (Warwick) |
| “Nonparametric inference for ratios of densities via uniformly valid and powerful permutation tests” (abstract) | |
| 5:15pm – 5:40pm | Marcel Ortgiese (Bath) |
| “The spatial Muller's ratchet” (abstract) | |
| 5:40pm – 5:50pm | Anastasia Kovtun (Cardiff) |
| “Multidimensional Dickman distribution and operator selfdecomposability” (abstract) | |
| 5:50pm – 6:00pm | Panqiu Xia (Cardiff) |
| “Ergodicity and Gaussian fluctuations of stochastic heat equations” (abstract) |
Tue 31st March, 2026
| 3:05pm – 3:30pm | Ellen Powell (Durham) |
| “Scaling limits of critical FK-decorated maps at \(q=4\)” (abstract) | |
| 3:35pm – 3:45pm | Zhengyan Wu (TU Munich) |
| “Fluctuating hydrodynamics in kinetic equations” (abstract) | |
| 3:45pm – 3:55pm | Ethan Baker (Birmingham) |
| “Polynomial ergodicity of the generalised relativistic Langevin equation” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 4:55pm | María Dolores Ruiz-Medina (Granada) |
| “Non-central limit theorems for sojourn measures of spatio-temporal random fields” (abstract) | |
| 4:55pm – 5:05pm | Nenad Šuvak (Osijek) |
| “Spectral representation of the transition density of killed inverse-gamma diffusion on a bounded state space” (abstract) | |
| 5:15pm – 5:25pm | Ivan Papić (Osijek) |
| “Stretched non-local Pearson diffusions” (abstract) | |
| 5:30pm – 5:55pm | William Fitzgerald (Manchester) |
| “Ordered random walks and the Airy line ensemble” (abstract) |
Wed 1st April, 2026
| 3:05pm – 3:30pm | Gesine Reinert (Oxford) |
| “Exponential random graph models analysed using Stein's method” (abstract) | |
| 3:35pm – 3:45pm | Ibrahim Kaddouri (Warwick) |
| “Late change-point in the preferential attachment random graph model” (abstract) | |
| 3:45pm – 3:55pm | Ollie Baker (Bristol) |
| “Entropy of soft random geometric graphs in general geometries” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 4:55pm | Domenico Marinucci (Rome Tor Vergata) |
| “The geometry of random neural networks” (abstract) | 4:55pm – 5:05pm | Leoni Carla Wirth (Oxford) |
| “Private synthetic graph generation” (abstract) | |
| 5:15pm – 5:25pm | Alexander Kent (Warwick) |
| “Rate optimality and phase transition for user-level local differential privacy” (abstract) | |
| 5:30pm – 5:55pm | Tom Berrett (Warwick) |
| “Permutation testing under local differential privacy” (abstract) |
Probability section organisers:
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Abstracts:
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Ethan Baker (Birmingham): Polynomial ergodicity of the generalised relativistic Langevin equation
In this talk, we will discuss the ergodicity of two relativistic Langevin equations. Relativistic Langevin equations describe the motion of a particle, with relativistic kinetic energy, subject to an external potential and a noise. We will first review existing results of the polynomial ergodicity of a Markov relativistic Langevin equation. We will then introduce a Relativistic Langevin Equation with memory, called the Generalised Relativistic Langevin Equation (GRLE), and re-formulate the equation as a Markov process by approximating the memory kernel as a sum of exponentials. With this formulation, we will construct a Lyapunov function for the GRLE, both in the presence and absence of friction, to obtain polynomial convergence rates of the transition densities to the invariant Gibbs measure.
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Ollie Baker (Bristol): Entropy of soft random geometric graphs in general geometries
Knowledge of the entropy of a random network is important in many areas, such as information theory and probability. When considering the Soft Random Geometric Graph, there is currently a lack of understanding of how the geometry in which the graph is embedded affects its information theoretic properties. In this work, we introduce the 'entropy graph' formalism, which allows us to study the amount of entropy that individual points contribute to the total entropy by studying their 'entropy mass'. We use this to prove a limit theorem in the small connection range limit, and quantify the effects of boundary components on the graph entropy in various geometries.
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Tom Berrett (Warwick): Permutation testing under local differential privacy
Personal and sensitive data is now collected at larger scales than ever before. Growing concern from data subjects and regulatory bodies, however, has led to an increased demand for statistical procedures that do not compromise the privacy of the individuals whose data are collected and analysed. In this talk I will discuss recent work on two-sample testing under a local differential privacy constraint where a permutation procedure is used to calibrate the tests. While permutation testing is a classical resampling technique, popular due to its ease of implementation and uniform Type I error control, its use under local privacy constraints is complicated by the fact that access to the data is limited. In this work we design appropriate privacy mechanisms, both interactive and non-interactive, that allow for permutation tests. Our analysis shows that these lead to minimax optimal separation rates in both discrete and continuous settings, with interactive procedures being significantly more powerful. This is recent joint work with Alexander Kent and Yi Yu.
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Alberto Bordino (Warwick): Nonparametric inference for ratios of densities via uniformly valid and powerful permutation tests
We propose the density ratio permutation test, a hypothesis test that assesses whether the ratio between two densities is proportional to a known function based on independent samples from each distribution. The test uses an efficient Markov Chain Monte Carlo scheme to draw weighted permutations of the pooled data, yielding exchangeable samples and finite sample validity. For power, if the statistic is an integral probability metric, our procedure is consistent under mild assumptions on the defining function class; specializing to a reproducing kernel Hilbert space, we introduce the shifted maximum mean discrepancy and prove minimax optimality of our test when a normalized difference between the densities lies in a Sobolev ball. We extend to the case of an unknown density ratio by estimating it on an independent training sample and derive type I error bounds in terms of the estimation error as well as power results. This allows adapting our method to conditional two sample testing, making it a versatile tool for assessing covariate-shift and related assumptions, which frequently arise in transfer learning and causal inference. Finally, we validate our theoretical findings through experiments on both simulated and real-world datasets.
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William Fitzgerald (Manchester): Ordered random walks and the Airy line ensemble
The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang (KPZ) universality class. In the KPZ universality class, it is related to the universal limiting objects that describe the scaling limit of a large class of random interface growth models and interacting particle systems. In random matrix theory, it is the edge scaling limit of Dyson Brownian motion, the evolution of the eigenvalues of Brownian motion in the space of Hermitian matrices. I will discuss these connections and a universality property of the Airy line ensemble. Consider a growing number of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. The top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power of the expected number of random walk steps.
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Gilles Germain (Oxford): Normal approximation of the intrinsic volumes distribution of a convex body
We introduce a new version of Stein's method of comparison of operators specifically tailored to the problem of bounding the Wasserstein-1 distance between continuous and discrete distributions on the real line. Our approach rests on a new family of weighted discrete derivative operators, which we call bespoke derivatives. The application that we have in mind concerns the convergence of the intrinsic volume distribution of a convex body to the normal distribution, a result that is currently conjectured. By means of our method, we prove the conjecture in the case of rectangular parallelotopes.
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Jessica Jay (Lancaster): Connections between interacting particle systems and combinatorial objects
In 2018 Balázs and Bowen gave a purely probabilistic proof to a well-known combinatorial identity, the Jacobi triple product identity. This identity links an infinite sum to an infinite triple product and has interpretations across Mathematics and Physics. Probabilistically the identity is given as an equivalence of reversible stationary measures for two classical particle systems via the Exclusion – Zero-range correspondence. Recent research has found other examples of probabilistic proofs to identities of combinatorial significance by studying natural questions for certain interacting particle systems. This connection is not only interesting but can be very useful for proving new and sometimes surprising results both in combinatorics and probability. In this talk we will see a selection of such results; new combinatorial identities and surprising probabilistic results which come from the connection with combinatorics. Based on a selection of works joint with Daniel Adams, Márton Balázs, Dan Fretwell and Benjamin Lees.
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Ibrahim Kaddouri (Warwick): Late change-point in the preferential attachment random graph model
We consider the problem of late change-point detection under the preferential attachment random graph model with time dependent attachment function. This can be formulated as a hypothesis testing problem where the null hypothesis corresponds to a preferential attachment model with a constant affine attachment parameter \(\delta_0\) and the alternative corresponds to a preferential attachment model where the affine attachment parameter changes from \(\delta_0\) to \(\delta_1\) at a time \(\tau_n = n - \Delta_n\) where \(0\leq \Delta_n \leq n\) and \(n\) is the size of the graph. It was conjectured in [BBCH23] that when observing only the unlabeled graph, detection of the change is not possible for \(\Delta_n = o(n^{1/2})\). In this work, we make a step towards proving the conjecture by proving the impossibility of detecting the change when \(\Delta_n = o(n^{1/3})\). We also study change-point detection in the case where the labeled graph is observed and show that change-point detection is possible if and only if \(\Delta_n \to \infty\), thereby exhibiting a strong difference between the two settings.
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Alexander Kent (Warwick): Rate optimality and phase transition for user-level local differential privacy
Given demands for rigorous data privacy guarantees from both a regulatory standpoint and from the concerns of the data subjects, definitions of privacy which can be theoretically validated are of great interest. One such method enjoying significant popularity in both academia and industry is that of differential privacy in which carefully calibrated noise is added to data to provide plausible deniability as to the true value.
Differential privacy appears in both the central model, where a trusted aggregator has access to the data and releases a privatised output, and the local model, where each user adds noise before publishing their (now privatised) data to a potentially untrusted aggregator.
Referring to the traditional setting where each of the n data subjects hold a single data point as item-level privacy, a growing field of interest is that of user-level privacy where each of the n users holds T observations and wishes to maintain the privacy of their entire collection. We consider the model of user-level local differential privacy, which is relatively unexplored. Indeed, even for a problem as fundamental as univariate mean estimation, prior to this work the minimax rate of estimation was undetermined.
We aim to fill this gap, obtaining minimax optimal estimation rates for a range of canonical statistical estimation problems including univariate and multidimensional mean estimation, sparse mean estimation, and non-parametric density estimation. We first derive a general minimax lower bound, which shows that the risk cannot, in general, be made to vanish for a fixed number of users even when T is arbitrarily large. We then derive matching, up to logarithmic factors, lower and upper bounds for the aforementioned canonical problems. In particular, with other model parameters held fixed, we observe phase transition phenomena in the minimax rates as T, the number of observations each user holds, varies.
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Anastasia Kovtun (Cardiff): Multidimensional Dickman distribution and operator selfdecomposability
The one-dimensional Dickman distribution arises in various stochastic models across number theory, combinatorics, physics, and biology. Recently, a definition of the multidimensional Dickman distribution has appeared in the literature, together with its application to approximating the small jumps of multidimensional Lévy processes. However, the newly defined distribution lacks the important property of operator selfdecomposability. Thus, we propose to extend this definition to a class of vector-valued random elements, which we characterise as fixed points of a specific affine transformation involving a random matrix obtained from the matrix exponential of a uniformly distributed random variable. We prove that these new distributions possess the key properties of infinite divisibility and operator selfdecomposability. Furthermore, we identify several cases where this new distribution arises as a limit. A preprint version of this work is available at arXiv:2602.12988.
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Nero Z. Li (Imperial): Iterated graph systems: Brownian motion on fractals
What is Brownian motion like on fractals? Classical studies of diffusions on special fractals, such as the Sierpiński gasket and the diamond hierarchical lattice, have laid much of the foundation for this question. Building on these developments, we introduce Iterated Graph Systems, a framework that generates a broad class of fractal graphs via substitutions. We prove that, under an appropriate rescaling, simple random walks on these graphs converge in the Gromov-Hausdorff-Prokhorov-Skorokhod topology to a diffusion on a compact metric measure space, which we regard as the associated Brownian motion. We further study the corresponding heat kernel behaviour, and introduce the degree dimension as a quantitative tool that unifies the locally finite and infinite-degree (scale-free) regimes within a single analytic framework.
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Domenico Marinucci (Rome Tor Vergata): The geometry of random neural networks
We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. More specifically, we show that, for activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. On the other hand, for activations which are more regular (e.g., ReLU, logistic and tanh), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. Our theoretical results are confirmed in some numerical experiments based on Monte Carlo simulations. Based on joint works with Simmaco Di Lillo, Michele Salvi and Stefano Vigogna.
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Marcel Ortgiese (Bath): The spatial Muller's ratchet
The spatial Muller’s ratchet is a spatial model of a population whose dynamics are shaped by the occurrence of deleterious mutations. The `ratchet’ refers to the effect that once a population has lost its fittest individuals it cannot recover these states. Mathematically, the model is described by a spatial birth-death process with rates depending on the local population density. Here, any additional mutation reduces the birth rate of an offspring. We show that under appropriate re-scaling, the process converges weakly to an infinite system of PDEs, confirming non-rigorous computations of Foutel-Rodier and Etheridge. Under certain conditions, we can analyse these PDEs and consider the behaviour of travelling waves exploring an empty habitat. Finally, we also answer the question whether deleterious mutations can surf population waves. Throughout we will discuss some of the technical difficulties that arise when dealing with a non-monotone particle system with infinitely many types and long-range interactions. This is joint work with Joao de Oliveira Madeira and Sarah Penington.
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Ivan Papić (Osijek): Stretched non-local Pearson diffusions
We introduce a class of stretched non-local Pearson diffusions obtained by replacing the classical first-order time derivative in the Kolmogorov equation of a Pearson diffusion with a stretched non-local time operator. The resulting processes retain the quadratic diffusion coefficient and linear drift structure characteristic of the Pearson family, while exhibiting non-Markovian temporal behaviour and anomalous scaling. We establish existence and uniqueness of solutions and derive explicit representations for transition densities via spectral expansions associated with the classical Pearson generator. The stretched time operator modifies the semigroup structure, yielding non-exponential relaxation with Kilbas–Saigo (Mittag–Leffler–type) asymptotics. These results provide a tractable framework connecting Pearson diffusions with non-local evolution equations, offering analytically explicit examples of diffusion models with memory while preserving the rich spectral structure of the underlying Pearson generator.
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Ellen Powell (Durham): Scaling limits of critical FK-decorated maps at
\(q=4\)
The critical Fortuin–Kasteleyn 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.
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Gesine Reinert (Oxford): Exponential random graph models analysed using Stein's method
Exponential random graph models are popular models for the analysis of social networks, due to their flexibility and to their ability of capturing some complex dependence in networks. Mathematically, their analysis is hindered by their probability distribution given only up to a normalising constant. Moreover usually only one observation from the network is available. Using ideas from Stein's method we can make progress though; we can characterize the model using a Stein operator and devise a kernelized Stein goodness-of-fit test based on this characterisation. We generate synthetic samples from the model underlying the observed data by mimicking the Stein operator dynamics. Finally, we show how a generalised approach can be used for anomaly detection.
This is based on joint works with Nathan Ross, Wenkai Xu, and Michal Kozyra.
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María Dolores Ruiz-Medina (Granada): Non-central limit theorems for sojourn measures of spatio-temporal random fields under spatio-temporal increasing domain asymptotics
This paper derives noncentral limit results (NCLTs) for sojourn measures of spatially homogeneous and isotropic, and stationary in time, LRD Chi-Squared Spatiotemporal Random Fields (STRFs). The cases of connected and compact two point homogeneous spaces \(\mathbb{M}_{d}\subset \mathbb{R}^{d+1}\), and compact convex sets \(\mathcal{K}\subset \mathbb{R}^{d+1}\) whose interior has positive Lebesgue measure, are analyzed. These NCLTs are obtained in the first Laguerre Chaos by applying reduction theorems. The methodological approaches adopted in the derivation of these results are based on the pure point and continuous spectra of the Gaussian STRFs generating the chi--squared subordinators defined on \(\mathbb{M}_{d}\) and \(\mathcal{K}\), respectively. Real-data applications where the results derived can be applied are discussed as well.
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Nenad Šuvak (Osijek): Spectral representation of the transition density of killed inverse-gamma diffusion on a bounded state space
A one-dimensional inverse-gamma diffusion taking values in \((0, K]\) with a space-dependent killing rate is investigated, where the boundary at \(K>0\) is assumed to be either absorbing or reflecting. By rewriting the infinitesimal generator of the killed diffusion in Sturm–Liouville form and imposing Dirichlet or Neumann boundary conditions at \(K\), it is shown that the associated operator possesses a purely discrete spectrum consisting of simple real eigenvalues and a complete orthogonal system of eigenfunctions. Using these spectral properties, a closed-form eigenfunction expansion for the transition probability density is derived. The obtained representation provides an analytical framework for studying transition behaviour and related functionals of inverse-gamma diffusions with boundary constraints and space-dependent killing.
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Tengyao Wang (LSE): Coverage correlation: detecting singular dependencies between random variables
We introduce the coverage correlation coefficient, a novel nonparametric measure of statistical association designed to quantifies the extent to which two random variables have a joint distribution concentrated on a singular subset with respect to the product of the marginals. Our correlation statistic consistently estimates an f-divergence between the joint distribution and the product of the marginals, which is \(0\) if and only if the variables are independent and \(1\) if and only if the copula is singular. Using Monge–Kantorovich ranks, the coverage correlation naturally extends to measure association between random vectors. It is distribution-free, admits an analytically tractable asymptotic null distribution, and can be computed efficiently, making it well-suited for detecting complex, potentially nonlinear associations in large-scale pairwise testing.
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Leoni Carla Wirth (Oxford): Private synthetic graph generation
Networks are a powerful tool to analyze complex data, for example from biology, finance, or the social sciences. Such data often contains sensitive information and therefore requires the application of mechanisms that ensure privacy of individuals. At the same time, it is crucial to preserve utility, that is, to guarantee that the transformed data retains important structural and population-level properties. Differential privacy provides a mathematical framework to formalize and analyze this trade-off between privacy and utility.
In this talk, we focus on differential privacy in the context of network generation. More precisely, starting with an input data set that is viewed as a collection of vertex attributes, we aim to construct a privatized network representation. To this end, we introduce a method that first applies a privacy mechanism to the set of vertex attributes and then jointly constructs network representations based on both the original and the privatized data. We show that the resulting procedure is differentially private and provide a theoretical analysis of its utility, both in expectation and in distribution. Finally, we illustrate the effectiveness of our approach through simulations.
This is based on joint work with Gholamali Aminian (Alan Turing Institute) and Gesine Reinert (University of Oxford).
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Zhengyan Wu (TU Munich): Fluctuating hydrodynamics in kinetic equations
We present a recent well-posedness result for the Vlasov-Fokker-Planck-Dean-Kawasaki (VFPDK) equation with correlated noise. This stochastic kinetic equation arises as a fluctuating mean-field limit of second-order Newtonian particle systems, bridging classical kinetic theory and stochastic fluctuation effects. The model features bounded nonlocal interactions together with a diffusion coefficient of square-root type, which introduces significant analytical difficulties. To overcome these issues, the analysis combines refined kinetic semigroup estimates with the framework of renormalized kinetic solutions.
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Panqiu Xia (Cardiff): Ergodicity and Gaussian fluctuations of stochastic heat equations
In this talk, I will discuss the ergodicity of the stochastic heat equation driven by centred Gaussian noise, which is white in time and coloured in space, satisfying the Dalang condition. I will also provide a sufficient condition for the ergodicity, and classify the invariant measures based on their expectations. Assuming the spatial correlation has a Riesz-type tail of the form \(|x|^{- \gamma}\), a Gaussian fluctuation result under diffusive scaling was established. Moreover, it has been confirmed that the diffusive scaling limit satisfies an Edwards–Wilkinson equation. This talk is based on joint work with Le Chen, Alex Dunlap, Cheng Ouyang, and Samy Tindel.
Contact:
All further questions and inquiries can be addressed to Kirstin Strokorb at:
- Ks3047 at bath dot ac dot uk