We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^{\beta}$ when $\beta>4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent"ghost"critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .
Binomial distributions capture the probabilities of `heads' outcomes when a (biased) coin is tossed multiple times. The coin may be identified with a distribution on the two-element set {0,1}, where the 1 outcome corresponds to `head'. One can also toss two separate coins, with different biases, in parallel and record the outcomes. This paper investigates a slightly different `bivariate' binomial distribution, where the two coins are dependent (also called: entangled, or entwined): the two-coin is a distribution on the product {0,1} x {0,1}. This bivariate binomial exists in the literature, with complicated formulations. Here we use the language of category theory to give a new succint formulation. This paper investigates, also in categorically inspired form, basic properties of these bivariate distributions, including their mean, variance and covariance, and their behaviour under convolution and under updating, in Laplace's rule of succession. Furthermore, it is shown how Expectation Maximisation works for these bivariate binomials, so that mixtures of bivariate binomials can be recognised in data. This paper concentrates on the bivariate case, but the binomial distributions may be generalised to the multivariate case, with multiple dimensions, in a straightforward manner.
In this paper, we examine the problem of sampling from log-concave distributions with (possibly) superlinear gradient growth under kinetic (underdamped) Langevin algorithms. Using a carefully tailored taming scheme, we propose two novel discretizations of the kinetic Langevin SDE, and we show that they are both contractive and satisfy a log-Sobolev inequality. Building on this, we establish a series of non-asymptotic bounds in $2$-Wasserstein distance between the law reached by each algorithm and the underlying target measure.
Liouville first passage percolation (LFPP) with parameter ξ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi > 0$$\end{document} is the family of random distance functions (metrics) (Dhϵ)ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D_h^{\epsilon })_{\epsilon > 0}$$\end{document} on C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document} obtained heuristically by integrating eξh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{\xi h}$$\end{document} along paths, where h is a variant of the Gaussian free field. There is a critical value ξcrit≈0.41\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _{\text {crit}} \approx 0.41$$\end{document} such that for ξ∈(0,ξcrit)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in (0, \xi _{\text {crit}})$$\end{document}, appropriately rescaled LFPP converges in probability uniformly on compact subsets of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document} to a limiting metric Dh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_h$$\end{document} on γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}-Liouville quantum gravity with γ=γ(ξ)∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \gamma (\xi ) \in (0,2)$$\end{document}. We show that the convergence is almost sure, giving an affirmative answer to a question posed by Gwynne and Miller (Invent. Math. 223, 213–333 (2021) [math.PR]).
Jason T. Fisher, Urša Ciuha, Leonidas G. Ioannou
et al.
AbstractGlobal warming has caused an increase in the frequency, duration, and intensity of summer heatwaves (HWs). Prolonged exposure to hot environments and orthostasis may cause conflicting demands of thermoregulation and blood pressure regulation on the vasomotor system, potentially contributing to cardiovascular complications and occupational heat strain. This study assessed cardiovascular and skin blood flow (SkBF) responses to orthostasis before, during and after a 3-day simulated HW. Seven male participants maintained a standard work/rest schedule for nine consecutive days split into three 3-day parts; thermoneutral pre-HW (25.4 °C), simulated HW (35.4 °C), thermoneutral post-HW. Gastrointestinal (Tgi) and skin (Tsk) temperatures, cardiovascular responses, and SkBF were monitored during 10-min supine and 10-min 60° head-up tilt (HUT). SkBF, indexed using proximal–distal skin temperature gradient (∆TskP-D), was validated using Laser-Doppler Flowmetry (LDF). The HW significantly increased heart rate, cardiac output and SkBF of the leg in supine; HUT increased SkBF of the arm and leg, and significantly affected all cardiovascular variables besides cardiac output. Significant regional differences in SkBF presented between the arm and leg in all conditions; the arm displaying vasodilation throughout, while the leg vasoconstricted in non-HW before shifting to vasodilation in the HW. Additionally, ∆TskP-D strongly correlated with LDF (r = −.78, p < 0.001). Prolonged HW exposure and orthostasis, individually, elicited significant changes in cardiovascular and SkBF variables. Additionally, varying regional blood flow responses were observed, suggesting the upper and lower vasculature receives differing vasomotor control. Combined cardiovascular alterations and shifts towards vasodilation indicate an increased challenge to industrial workers during HWs.
We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.
Vladimir Anisimov, Guillaume Mijoule, Armando Turchetta
et al.
This paper focuses on statistical modelling and prediction of patient recruitment in clinical trials accounting for patients dropout. The recruitment model is based on a Poisson-gamma model introduced by Anisimov and Fedorov (2007), where the patients arrive at different centres according to Poisson processes with rates viewed as gamma-distributed random variables. Each patient can drop the study during some screening period. Managing the dropout process is of a major importance but data related to dropout are rarely correctly collected. In this paper, a few models of dropout are proposed. The technique for estimating parameters and predicting the number of recruited patients over time and the recruitment time is developed. Simulation results confirm the applicability of the technique and thus, the necessity to account for patients dropout at the stage of forecasting recruitment in clinical trials.
We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^3$$\end{document}. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^3$$\end{document} by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Φ34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^4_3$$\end{document}-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.
We propose alternatives to Bayesian a priori distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well defined infinite-dimensional random variables, and can be approximated by finite-dimensional random variables. We introduce a new wavelet-based model, where the non zero coefficient are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in some Besov spaces and have singularities only on a small set $τ$ that has a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in denoising problem.
Spin q-Whittaker symmetric polynomials labeled by partitions λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} were recently introduced by Borodin and Wheeler (Spin q-Whittaker Polynomials, 2017. arXiv preprint arXiv:1701.06292 [math.CO]) in the context of integrable sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {sl}_2$$\end{document} vertex models. They are a one-parameter deformation of the t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document} Macdonald polynomials. We present a new more convenient modification of spin q-Whittaker polynomials and find two Macdonald type q-difference operators acting diagonally in these polynomials with eigenvalues, respectively, q-λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{-\lambda _1}$$\end{document} and qλN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{\lambda _N}$$\end{document} (where λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is the polynomial’s label). We study probability measures on interlacing arrays based on spin q-Whittaker polynomials, and match their observables with known stochastic particle systems such as the q-Hahn TASEP. In a scaling limit as q↗1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\nearrow 1$$\end{document}, spin q-Whittaker polynomials turn into a new one-parameter deformation of the gln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_n$$\end{document} Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as q↗1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\nearrow 1$$\end{document} we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2016. arXiv:1503.04117 [math.PR]), and relate it to spin Whittaker functions.
Lions (1959, Bull. Soc. Math. France, \textbf{87}, 245--273) introduced the Navier-Stokes equations with a viscous diffusion in the form of a fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed the uniqueness of its solution when its exponent is not less than five quarters in case the spatial dimension is three. Following the work of Hofmanov$\acute{\mathrm{a}}$, Zhu and Zhu (2019, arXiv:1912.11841 [math.PR]), we prove the non-uniqueness in law for the three-dimensional stochastic Navier-Stokes equations with the viscous diffusion in the form of a fractional Laplacian with its exponent less than five quarters.
Many integrable stochastic particle systems in one space dimension (such as TASEP - Totally Asymmetric Simple Exclusion Process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations (namely, $q$-TASEP and beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution. Setting $q=0$, we recover the results about the usual TASEP established recently in arXiv:1907.09155 [math.PR] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.