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We consider a large class of deformations of continuous and discrete biorthogonal ensembles and investigate their behavior in the limit of a large number of particles. We provide sufficient conditions to ensure that if a biorthogonal ensemble converges to a (universal) limiting process, then the deformed biorthogonal ensemble converges to a deformed version of the same limiting process. To construct the deformed version of the limiting process, we rely on a procedure of marking and conditioning. Our approach is based on an analysis of the probability generating functionals of the ensembles and is conceptually different from the traditional approach via correlation kernels. Thanks to this method, our sufficient conditions are rather mild and do not rely on much regularity of the original ensemble and of the deformation. As a consequence of our results, we obtain probabilistic interpretations of several Painlev\'e-type kernels that have been constructed in the literature, as deformations of classical sine and Airy point processes., Comment: 32 pages

We present a general framework for constructing solvable lattice models of chaotic many-body quantum dynamics with multiple unitary directions using biunitary connections. We show that a network of biunitary connections on the Kagome lattice naturally defines a multi-unitary circuit, where three `arrows of time' directly reflect the lattice symmetry. These models unify various constructions of hierarchical dual-unitary and triunitary gates and present new families of models with solvable correlations and entanglement dynamics. Using multilayer constructions of biunitary connections, we additionally introduce multilayer circuits with monoclinic symmetry and higher level hierarchical dual-unitary solvability and discuss their (non-)ergodicity. Our work demonstrates how different classes of solvable models can be understood as arising from different geometric structures in spacetime., Comment: 34 pages, 1 figure

We conjecture an explicit expression for the lower tail large deviation rate function of the partition function of the log-Gamma polymer. We rigorously prove our result, except for one step for which we only provide heuristic evidence. We show that the large deviation rate function matches with that of last passage percolation with exponential weights in the zero-temperature limit, and with the lower tail of the Tracy-Widom distribution for moderate deviations., Comment: 40 pages, 7 figures

The chaotic dynamics of quantum many-body systems are expected to quickly randomize any structured initial state, delocalizing it in Fock space. In this work, we study the spreading of an initial product state in Hilbert space under dual-unitary dynamics, captured by the inverse participation ratios and the distribution of overlaps (bit-string probabilities). We consider the self-dual kicked Ising model, a minimal model of many-body quantum chaos which can be seen as either a periodically driven Floquet model or a dual-unitary quantum circuit. Both analytically and numerically, we show that the inverse participation ratios rapidly approach their ergodic values, corresponding to those of Haar random states, and establish the emergence of the Porter-Thomas distribution for the overlap distribution. Importantly, this convergence happens exponentially fast in time, with a time scale that is independent of system size. We inspect the effect of local perturbations that break dual-unitarity and show a slowdown of the spreading in Fock space, indicating that dual-unitary circuits are maximally efficient at preparing random states., Comment: 6+2 pages

Many-body quantum dynamics defined on a spatial lattice and in discrete time -- either as stroboscopic Floquet systems or quantum circuits -- has been an active area of research for several years. Being discrete in space and time, a natural question arises: when can such a model be viewed as evolving unitarily in space as well as in time? Models with this property, which sometimes goes by the name space-time duality, have been shown to have a number of interesting features related to entanglement growth and correlations. One natural way in which the property arises in the context of (brickwork) quantum circuits is by choosing dual unitary gates: two site operators that are unitary in both the space and time directions. We introduce a class of models with $q$ states per site, defined on the square lattice by a complex partition function and paremeterized in terms of $q\times q$ Hadamard matrices, that have the property of space-time duality. These may interpreted as particular dual unitary circuits or stroboscopically evolving systems, and generalize the well studied self-dual kicked Ising model. We explore the operator dynamics in the case of Clifford circuits, making connections to Clifford cellular automata [J. Math. Phys. 49, 112104 (2008)] and in the $q\to\infty$ limit to the classical spatiotemporal cat model of many body chaos [Nonlinearity 34, 2800 (2021)]. We establish integrability and the corresponding conserved charges for a large subfamily and show how the long-range entanglement protocol discussed in the recent paper [Phys. Rev. B 105, 144306 (2022)] can be reinterpreted in purely graphical terms and directly applied here.

The use of Bluetooth Low Energy in low-range Internet of Things systems is growing exponentially. Similar to other wireless communication protocols, throughput and reliability are two key performance metrics in Bluetooth Low Energy communications. However, electromagnetic interference from various sources can heavily affect the performance of wireless devices, leading to dropped throughput and unreliable communication. Therefore, there is a need for both theoretical and practical studies capable of quantifying the BLE communication performance, e.g. throughput and reliability, subject to interference. In this paper, a mathematical model to predict throughput of a BLE connection under interference is derived first, and linked to the reliability model we developed in [1]. After that, extensive practical experiments are performed in various scenarios to sufficiently validate the theoretical results from both models. Finally, the trade-off between throughput and reliability is investigated through the validated models to give some inside properties of BLE communications. The similarity between the theoretical results and the experimental ones highlights the accuracy of the proposed throughput and reliability models. Hence, the two models can be used to explore the performance of various BLE designs or deployments from diverse perspectives.

We study temporal entanglement in dual-unitary Clifford circuits with probabilistic measurements preserving spatial unitarity. We exactly characterize the temporal entanglement barrier in the measurement-free regime, exhibiting ballistic growth and decay and a volume-law peak. In the presence of measurements, we relate the temporal entanglement to the scrambling properties of the circuit. For "good scramblers" measurements do not qualitatively change the temporal entanglement profile but only result in a reduced entanglement velocity, whereas for "poor scramblers" the initial ballistic growth of temporal entanglement with bath size is modified to diffusive. This difference is understood through a mapping of the underlying operator dynamics to a biased and an unbiased persistent random walk respectively. In the latter case measurements additionally modify the ballistic decay to the perfect dephaser limit, with vanishing temporal entanglement, to an exponential decay, which we describe through a spatial transfer matrix method. This spatial dynamics is shown to be described by a non-Hermitian hopping model, exhibiting a PT-breaking transition at a critical measurement rate $p=1/2$. In all cases the peak value of the temporal entanglement barrier exhibits volume-law scaling for all measurement rates.

Mitral Transcatheter Edge-to-Edge Repair (mTEER) is a medical procedure utilized for the treatment of mitral valve disorders. However, predicting the outcome of the procedure poses a significant challenge. This paper makes the first attempt to harness classical machine learning (ML) and deep learning (DL) techniques for predicting mitral valve mTEER surgery outcomes. To achieve this, we compiled a dataset from 467 patients, encompassing labeled echocardiogram videos and patient reports containing Transesophageal Echocardiography (TEE) measurements detailing Mitral Valve Repair (MVR) treatment outcomes. Leveraging this dataset, we conducted a benchmark evaluation of six ML algorithms and two DL models. The results underscore the potential of ML and DL in predicting mTEER surgery outcomes, providing insight for future investigation and advancements in this domain., Comment: 5 pages, 1 figure

We develop the study of a particular class of biorthogonal measures, encompassing at the same time several random matrix models and partition functions of polymers. This general framework allows us to characterize the partition functions of the Log Gamma polymer and the mixed polymer in terms of explicit biorthogonal measures, as it was previously done for the homogeneous O'Connell-Yor polymer by Imamura and Sasamoto. In addition, we show that the biorthogonal measures associated to these three polymer models (Log Gamma, O'Connell-Yor, and mixed polymer) converge to random matrix eigenvalue distributions in small temperature limits. We also clarify the connection between different Fredholm determinant representations and explain how our results might be useful for asymptotic analysis and large deviation estimates., Comment: 48 pages, 3 figures, 1 table. Minor modifications in section 7. Final version accepted for publication in Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques

The generalized optimised Schwarz method proposed in [Claeys & Parolin, 2022] is a variant of the Despr\'es algorithm for solving harmonic wave problems where transmission conditions are enforced by means of a non-local exchange operator. We introduce and analyse an acceleration technique that significantly reduces the cost of applying this exchange operator without deteriorating the precision and convergence speed of the overall domain decomposition algorithm.

Entanglement membrane theory is an effective coarse-grained description of entanglement dynamics and operator growth in chaotic quantum many-body systems. The fundamental quantity characterizing the membrane is the entanglement line tension. However, determining the entanglement line tension for microscopic models is in general exponentially difficult. We compute the entanglement line tension in a recently introduced class of exactly solvable yet chaotic unitary circuits, so-called generalized dual-unitary circuits, obtaining a non-trivial form that gives rise to a hierarchy of velocity scales with $v_E

The Tavis-Cummings model is a paradigmatic central-mode model where a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we here establish the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates and illustrates how these models support a collective spin description despite the presence of disorder., Comment: 10+6 pages

The Hayden-Preskill protocol probes the capability of information recovery from local subsystems after unitary dynamics. As such it resolves the capability of quantum many-body systems to dynamically implement a quantum error-correcting code. The transition to coding behavior has been mostly discussed using effective approaches, such as entanglement membrane theory. Here, we present exact results on the use of Hayden-Preskill recovery as a dynamical probe of scrambling in local quantum many-body systems. We investigate certain classes of unitary circuit models, both structured Floquet (dual-unitary) and Haar-random circuits. We discuss different dynamical signatures corresponding to information transport or scrambling, respectively, that go beyond effective approaches. Surprisingly, certain chaotic circuits transport information with perfect fidelity. In integrable dual-unitary circuits, we relate the information transmission to the propagation and scattering of quasiparticles. Using numerical and analytical insights, we argue that the qualitative features of information recovery extend away from these solvable points. Our results suggest that information recovery protocols can serve to distinguish chaotic and integrable behavior, and that they are sensitive to characteristic dynamical features, such as long-lived quasiparticles or dual-unitarity., Comment: 28 pages, 10 figures

Dispersion diagrams play a crucial role in examining, analyzing and designing wave propagation in periodic structures. Despite their ubiquity and current research interest, introductory papers and reference scripting tailored to novel researchers in the field are lacking. This paper aims to address this gap, by presenting a comprehensive educational resource for researchers starting in the field of periodic structures and more specifically on the study of dispersion relations captured by dispersion surfaces or curves in dispersion diagrams. The objective is twofold. A first objective is to give a detailed explanation of dispersion diagrams, with graphical illustrations. Secondly, a documented Matlab code is provided to compute dispersion curves of 3D structures with 2D periodicity using the so-called inverse approach. These dispersion curves are obtained with numerical simulations using the finite element method. The code is written for elastic wave propagation and orthogonal periodicity directions, but can be extended to other types of linear wave propagation, non-orthogonal periodicity directions or 1D and 3D periodicity. The aim of this code is to serve as a starting point for novice researchers in the field, to facilitate their understanding of different aspects of dispersion diagrams and serve as a stepping stone in their future research., Comment: 43 pages, 23 figures

We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a $2$-dimensional triangulated surface $\Gamma$ in $\mathbb{R}^3$. We allow $\Gamma$ to belong to a large class of geometries that we call polygonal multiscreens, which can be non-manifold. After introducing a new, simple conforming Galerkin discretization, we analyze a substructuring domain-decomposition preconditioner based on ideas originally developed for the Finite Element Method. The surface $\Gamma$ is subdivided into non-overlapping regions, and the application of the preconditioner is obtained via the solution of the hypersingular equation on each patch, plus a coarse subspace correction. We prove that the condition number of the preconditioned linear system grows poly-logarithmically with $H/h$, the ratio of the coarse mesh and fine mesh size, and our numerical results indicate that this bound is sharp. This domain-decomposition algorithm therefore guarantees significant speedups for iterative solvers, even when a large number of subdomains is used.

We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We establish a connection between these determinants and a system of integro-differential equations generalizing the fifth Painlev\'e equation, and we show that they allow us to solve an integrable PDE explicitly for a large class of initial data., Comment: 26 pages

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ is uniformly of $d$-dimensional Hausdorff dimension in a sense we make precise (a $d$-set), the operator in our equation is an integral operator on $\Gamma$ with respect to $d$-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code.

We investigate many-body dynamics where the evolution is governed by unitary circuits through the lens of `Krylov complexity', a recently proposed measure of complexity and quantum chaos. We extend the formalism of Krylov complexity to unitary circuit dynamics and focus on Floquet circuits arising as the Trotter decomposition of Hamiltonian dynamics. For short Trotter steps the results from Hamiltonian dynamics are recovered, whereas a large Trotter step results in different universal behavior characterized by the existence of local maximally ergodic operators: operators with vanishing autocorrelation functions, as exemplified in dual-unitary circuits. These operators exhibit maximal complexity growth, act as a memoryless bath for the dynamics, and can be directly probed in current quantum computing setups. These two regimes are separated by a crossover in chaotic systems. Conversely, we find that free integrable systems exhibit a nonanalytic transition between these different regimes, where maximally ergodic operators appear at a critical Trotter step., Comment: 22 pages, 16 figures

Considering positive Maxwell problems, we propose a substructured version of the Hiptmair-Xu preconditioner based on a new formula that expresses the inverse of Schur systems in terms of the inverse matrix of the global volume problem.

We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a G{\aa}rding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.

This report serves as a technology description of a Julia-based re-implementation of the fixed-point current injection algorithm, available in PowerModelsDistribution.jl [1]. This report does not describe a novel method for solving unbalanced power flow problems. It merely provides a description of the fixed point iteration variant of the current injection method, inspired by the existing open-source implementation in OpenDSS1 [2]. The current injection method is commonly conceived as a system of nonlinear equalities solved by Newton s method [3, 4]. However, as Roger Dugan points out in the OpenDSS documentation, the fixed point iteration variant commonly outperforms most methods, while supporting meshed topologies from the ground up. We note that the unbalanced power flow algorithm in turn relies on matrix solvers for sparse systems of equations. In the context of circuits and factorizing nodal admittance matrices, the sparsity-exploiting KLU solver [5] has proven to be both reliable and scalable. OpenDSS uses KLU. This report documents work-in-progress, and the authors aim to update it when learnings are obtained or more features are added to the implementation in PowerModelsDistribution.jl. The authors invite collaborators to contribute through pull requests on the repository.

We study J\'anossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg-de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of Amir-Corwin-Quastel's integro-differential Painlev\'e II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg-de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions., Comment: V1: 41 pages, 3 figures. V2: 45 pages, 3 figures; revised introduction, minor corrections. V3: 46 pages, 3 figures; minor corrections

Dual-unitary brickwork circuits are an exactly-solvable model for many-body chaotic quantum systems, based on 2-site gates which are unitary in both the time and space directions. Prosen has recently described an alternative model called 'dual-unitary interactions round-a-face', which we here call 'clockwork', based on 2-controlled 1-site unitaries composed in a non-brickwork structure, yet with many of the same attractive global properties. We present a 2-categorical framework that simultaneously generalizes these two existing models, and use it to show that brickwork and clockwork circuits can interact richly, yielding new types of generalized heterogeneous circuits. We show that these interactions are governed by quantum combinatorial data, which we precisely characterize. These generalized circuits remain exactly-solvable and we show that they retain the attractive features of the original models such as single-site correlation functions vanishing everywhere except on the causal light-cone. Our presented framework allows us to directly extend the notion of solvable initial states to these biunitary circuits, which are shown to result in maximal entanglement growth and exact thermalization after finitely many time steps under biunitary circuit dynamics., Comment: 29 pages. Updated introduction to shaded calculus, introduced periodic biunitary circuits in Section 2.5, and provided explicit parametrizations of composite biunitaries in Appendix A

We extend the theoretical framework of non-local optimized Schwarz methods as introduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries. The problem is reformulated equivalently as an equation posed on the skeleton of a non-overlapping partition of the computational domain, involving an operator of the form "identity + contraction". The analysis covers the possibility of resonance phenomena where the Helmholtz problem is not uniquely solvable. In case of unique solvability, the skeleton formulation is proved coercive, and an explicit bound for the coercivity constant is provided in terms of the inf-sup constant of the primary Helmholtz boundary value problem.

This work introduces ``generalized meshes", a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend regular simplicial meshes by allowing for overlapping elements and more flexible adjacency relations. They can have several distinct ``generalized" vertices (or edges, faces) that occupy the same geometric position. These generalized facets are the natural degrees of freedom for classical conforming spaces of discrete differential forms appearing in finite and boundary element applications. Special attention is devoted to the representation of fractured domains and their boundaries. An algorithm is proposed to construct the so-called {\em virtually inflated mesh}, which correspond to a ``two-sided" mesh of a fracture. Discrete $d$-differential forms on the virtually inflated mesh are characterized as the trace space of discrete $d$-differential forms in the surrounding volume.

Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-$s$ central spin XX model is integrable for any $s$., Comment: 38 pages, 11 figures; (v2) minor corrections; (v3) additional level spacing data for higher spin models; (v4) minor corrections

We investigate the influence of the choice of the partition of unity on the convergence of the Symmetrized Optimized Restricted Additive Schwarz (SORAS) preconditioner for the reaction-convection-diffusion equation. We focus on two kinds of partitions of unity, and study the dependence on the overlap and on the number of subdomains. In particular, the second kind of partition of unity, which is non-zero in the interior of the whole overlapping region, gives more favorable convergence properties, especially when increasing the overlap width, in comparison with the first kind of partition of unity, whose gradient is zero on the subdomain interfaces and which would be the natural choice for ORAS solver instead.

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of non-generic behaviour. By investigating the effect of weakly broken dual-unitarity on the spreading of local operators we study whether, and how, small deviations from dual-unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and use it to recover a butterfly velocity smaller than the light-cone velocity, $v_B < v_{LC}$ , and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. We find that the butterfly velocity and diffusion constant are determined by a small set of microscopic quantities and that the operator entanglement of the gates plays a crucial role., Comment: (6+15) pages, 5 figures; Accepted version; corrected some typos in the supplementary material

The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight $\exp(-n |x|^{\beta})$, $\beta>0$, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in $\beta$. If $\beta\ge 1$, it is described by the standard sine process. Below the critical value $\beta=1$, it is described by a process depending on the value of $\beta$, and we determine the first two terms of the large gap probability in it. This so called weak confinement range $0<\beta<1$ corresponds to the Freud weight with the indeterminate moment problem. We also find the multiplicative constant in the asymptotic expansion of the Freud multiple integral for $\beta\ge 1$., Comment: 56 pages

Using asymptotics of Toeplitz+Hankel determinants, we establish formulae for the asymptotics of the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices, as the matrix-size tends to infinity. Our results are analogous to those that Fahs obtained for random unitary matrices in [14]. A key feature of the formulae we derive is that the phase transitions in the moments of moments are seen to depend on the symmetry group in question in a significant way.