Stochastic quantisation of Yang–Mills

We review two works arXiv:2006.04987 and arXiv:2201.03487 which study the stochastic quantisation equations of Yang-Mills on two and three dimensional Euclidean space with finite volume. The main result of these works is that one can renormalise the 2D and 3D stochastic Yang-Mills heat flow so that the dynamic becomes gauge covariant in law. Furthermore, there is a state space of distributional $1$-forms $\mathcal{S}$ to which gauge equivalence $\sim$ extends and such that the renormalised stochastic Yang-Mills heat flow projects to a Markov process on the quotient space of gauge orbits $\mathcal{S}/{\sim}$. In this review, we give unified statements of the main results of these works, highlight differences in the methods, and point out a number of open problems.Comment: 32 pages, 1 figure. Fixed typos and updated references. To appear in Journal of Mathematical Physics for the proceedings of ICMP 202

On the Number of Facets of Polytopes Representing Comparative Probability Orders

Fine and Gill (1973) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by a region of a certain hyperplane arrangement. Maclagan (1999) asked how many facets a polytope, which is the closure of such a region, might have. We prove that the maximal number of facets is at least F_{n+1}, where F_n is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our proof is combinatorial and makes use of the concept of flippable pairs introduced by Maclagan. We also obtain an upper bound which is not too far from the lower bound.Comment: 13 page

Signature moments to characterize laws of stochastic processes

The normalized sequence of moments characterizes the law of any finite-dimensional random variable. We prove an analogous result for path-valued random variables, that is stochastic processes, by using the normalized sequence of signature moments. We use this to define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic processes.Comment: 31 pages, 5 figure

Gauge field marginal of an Abelian Higgs model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon--Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler but holds in greater generality. Our second main result shows ultraviolet stability for the gauge field marginal in a fixed gauge. More specifically, we show that moments of the H\"older--Besov-type norms introduced in arXiv:1808.09196 are bounded uniformly in the lattice spacing. This second result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.Comment: 34 pages, 6 figure

Invariant measure and universality of the 2D Yang-Mills Langevin dynamic

We prove that the Yang-Mills (YM) measure for the trivial principal bundle over the two-dimensional torus, with any connected, compact structure group, is invariant for the associated renormalised Langevin dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing, and Bourgain's method for invariant measures. Several corollaries are presented including a gauge-fixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder, and a proof of universality for the YM measure that we derive from a universality for the Langevin dynamic for a wide class of discrete approximations. The latter includes standard lattice gauge theories associated to Wilson, Villain, and Manton actions. An important step in the argument, which is of independent interest, is a proof of uniqueness for the mass renormalisation of the gauge-covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. This latter result relies on Euler estimates for singular SPDEs and for Young ODEs arising from Wilson loops.Comment: 157 pages. Shortened the earlier version. Strengthened uniqueness result in Sec 8 which allows simplifications in Sec 3 and 5.1. Simplified Sec 5.5. Minor corrections elsewhere in the pape

Constructing supersingular elliptic curves with a given endomorphism ring

Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice O^T of O, the order O is effectively characterized by the three successive minima and two other short vectors of O^T. The desired conditions turn out to hold whenever the j-invariant j(E), of the elliptic curve with End(E) = O, lies in F_p. We can then prove that our algorithm terminates with running time O(p^{1+\epsilon}) under the aforementioned conditions. As a further application we present an algorithm to simultaneously match all maximal order types with their associated j-invariants. Our algorithm has running time O(p^{2.5+\epsilon}) operations and is more efficient than Cervino's algorithm for the same problem.Comment: Full version of paper published by the LMS Journal of Computation and Mathematic

A support and density theorem for Markovian rough paths

We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X}$ in $\alpha$-H\"older rough path topology for all $\alpha \in (0,1/2)$, which answers in the positive a conjecture of Friz-Victoir (2010). The second is a H\"ormander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X}$, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.Comment: 17 pages. Added several clarifications. To appear in Electron. J. Proba

Characteristic functions of measures on geometric rough paths

We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from L\'evy, Gaussian and Markovian rough paths.Comment: 29 pages, published version, updated ref

An isomorphism between branched and geometric rough paths

We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly. Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths.Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths

A Primer on the Signature Method in Machine Learning

In these notes, we wish to provide an introduction to the signature method, focusing on its basic theoretical properties and recent numerical applications. The notes are split into two parts. The first part focuses on the definition and fundamental properties of the signature of a path, or the path signature. We have aimed for a minimalistic approach, assuming only familiarity with classical real analysis and integration theory, and supplementing theory with straightforward examples. We have chosen to focus in detail on the principle properties of the signature which we believe are fundamental to understanding its role in applications. We also present an informal discussion on some of its deeper properties and briefly mention the role of the signature in rough paths theory, which we hope could serve as a light introduction to rough paths for the interested reader. The second part of these notes discusses practical applications of the path signature to the area of machine learning. The signature approach represents a non-parametric way for extraction of characteristic features from data. The data are converted into a multi-dimensional path by means of various embedding algorithms and then processed for computation of individual terms of the signature which summarise certain information contained in the data. The signature thus transforms raw data into a set of features which are used in machine learning tasks. We will review current progress in applications of signatures to machine learning problems.Comment: 45 pages, 25 figure