We prove that deterministic motion in dissipative systems emerges as a strict geometric attractor of contact flow, not a statistical approximation. Building on the contact geometry of stochastic vector bundles, we develop time-dependent contact potentials with an exact closure theorem ensuring exact satisfaction of the master equation at any finite order. The Contact Locking Theorem shows that exponential gradient amplification of the probability field is precisely counterbalanced by synchronous stiffness decay, forcing the effective macroscopic-microscopic coupling to vanish exponentially. Deterministic dynamics therefore emerges through deterministic focusing with a universal timescale governed by the drift-field Jacobian spectrum. Validation of the damped-driven Duffing oscillator confirms the predicted rate and exponential convergence.

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian, describing a two-state quantum system linearly coupled to a bosonic quantum field. Similar to recent FKFs for the Fröhlich polaron and the non- and semi-relativistic Nelson models, it yields a probabilistic treatment of the spin as a jump process, but treats the field on the usual bosonic Fock space. As an application, we prove that the existence of ground states for infrared-regular models persists the removal of an ultraviolet cutoff.

Advanced adaptive optics (AO) systems can use Fourier-type wavefront sensing to correct optical distortions encountered in ground-based telescopes, AO-assisted retinal imaging, and free-space optical communications (FSOC). Recently, a novel Fourier-type wavefront sensor (WFS) known as the iQuad WFS has been introduced. Its design features a focal plane tessellation with a four-quadrant phase mask (FQPM) that incorporates a $\pm \pi/2$ phase shift between adjacent quadrants. In this work, we establish a comprehensive mathematical framework for the iQuad WFS, including its forward models and linearizations based on the Fréchet derivative. We reveal a connection between the iQuad WFS and the 2d finite Hilbert transform and demonstrate that the linear iQuad WFS operator is self-adjoint - a unique property among Fourier-type WFSs. Additionally, we introduce the double iQuad WFS, a two-path configuration that combines two rotated iQuad WFSs. This design addresses the limitations of the single iQuad WFS by suppressing poorly-seen phase components. Moreover, the double setup simplifies the mathematical modeling. We also highlight iQuad similarities to the widely used pyramid wavefront sensor (PWFS). Finally, we extend the concept of modulation to the iQuad WFS, further enhancing its versatility. The theoretical analysis presented here lays the groundwork for the development of fast and robust model-based wavefront reconstruction algorithms for the iQuad WFS, paving the way for future applications in AO instruments.

We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.

Land reclamation methods, indispensable for the proper development of modern coastal cities, are ecologically destructive. We present a fractal structure, similar to a Sierpinski triangle, which solves this problem by resting directly on the seabed thanks to the uniform load distribution we achieve on its base. To obtain this uniform distribution, we show that the supports of the structure must displace vertically following any function of the Takagi class. This causes the vertical deformations of the structure to follow this same class and the horizontal deformations to be related to the Cantor function. The structure works with an unlimited number of combinations of areas of its elements and materials, which gives designers a high degree of constructive flexibility.

A Wiener-Hopf-type integral equation for the stationary PDF of a reflected random walk is derived rigorously based on modern probability theory, and an application to battery energy storage systems (BESS), specifically the sizing of the inverter, is discussed in depth. The methodological steps include the construction of a Markov kernel, the derivation of a Fredholm integral equation of the second kind for the PDF of the BESS power, and an analytical solution of the equation based on a Neumann series. The analytical results were compared against numerical solutions obtained with the Nystrom method, as well as against the results of an algorithmic simulation using simulated input time series. The use of truncated versions of the analytic solution allows for the construction of simplified design rules for the power systems practitioner. General insights into inverter sizing criteria of storage systems for ramp-rate control of variable renewable energy (VRE) sources such as wind and solar are provided.

It was recently proven that, in contrast to their matrix analogues, the moments of a real Gaussian tensor of size N do not in general factorize over their connected components in the asymptotic large N limit. While the original proof of this rather surprising result was not constructive, explicit examples of non-factorizing moments, which are expectation values of trace-invariants, have since then been discovered. We explore further aspects of this problem, with a focus on Haar-distributed (or Gaussian) complex random tensors, which are more directly relevant to quantum information. We start out by exhibiting an explicit example of non-factorizing trace-invariant, thereby filling a gap in the recent literature. We then turn to the opposite question: that of finding interesting families of trace-invariants that do in fact factorize at large N. We establish three main theorems in this regard. The first one provides a sufficient combinatorial bound ensuring large N factorization, that is also simple enough to be applicable to various cases of practical relevance. Our second main result shows that the expectation value of any compatible trace-invariant is dominated by certain tree-like combinatorial structures at large N, which we refer to as tree-like dominant pairings. Our third main theorem establishes that any trace-invariant admitting tree-like dominant pairings does actually factorize at large N. In this way, we are able to prove that various families of trace-invariants that have been previously studied in the literature do factorize at large N. We apply our findings to the theory of multipartite quantum entanglement: to any trace-invariant is associated a multipartite generalization of Rényi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.

We study lower bounds for the quantum relative entropy between a density matrix and its block-diagonal part. For a block matrix with diagonal blocks A,C>0 and off-diagonal coherence block B, we prove a lower bound expressed through the associated Bogoliubov--Kubo--Mori (BKM) kernel. The proof uses a midpoint estimate for the BKM Hessian along the affine interpolation between the matrix and its block-diagonal projection. The resulting estimate is genuinely noncommutative and retains information about the joint spectral structure of the diagonal blocks and the coherence term. As a consequence, under a spectral gap condition on A relative to C, we obtain an explicit logarithmic lower bound proportional to the squared Frobenius norm of the coherence block. The appearance of the BKM metric is natural in this setting because it coincides with the Hessian of quantum relative entropy.

Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$ to $J(X)$, lifting the Abel map to $J(\widetilde X)$. We build a theta function on a compactification of the generalised Jacobian $J(X)$ (giving a section of a suitable positive line bundle). The translation action on $J(X)$ then yields all line bundles of that degree, and the translates of the theta function, restricted to $A(\widetilde X)$, give a ``universal section'' of the line bundles of that degree over $X$. This extends to the singular case a classical result of Riemann.

The goal of this article is twofold. We introduce a notion of convergence for Lorentzian pre-length spaces, $\ell$-convergence, that extends previous convergence notions in this context. We show that timelike curvature and timelike curvature-dimension bounds are stable under (measured) $\ell$-convergence. Then, we show that $\ell$-convergence is well adapted for generalized Lorentzian cones: a sequence of generalized cones $-I_i\times_{f_i}X_i$ converges in $\ell$ sense if the base $I_i$ and the fiber $X_i$ converge in GH sense and the functions $f_i$ converge uniformly. We use this to show sharp timelike curvature and timelike curvature-dimension bounds for such cones. Finally, we obtain a pre-compactness theorem for $\ell$-convergence in the class of smooth generalized cones that have a uniform lower bound on the full Ricci (or Riemann) curvature tensor.

We consider the defocusing nonlinear Schr{ö}dinger equation in the energy-subcritical case, and investigate the dependence of the solution upon the power of the nonlinearity. Special attention is paid to the global in time description. The main three aspects addressed, in the decreasing order of difficulty, are the limit when the total power tends to one, along with the connection with the logarithmic Schr{ö}dinger equation, the description when long range effects may be present, and the continuity of the scattering operator in the short range case. This text resumes the presentation given by the first author at {É}cole polytechnique for the Laurent Schwartz seminar, in May 2026.

Research on topological phases of matter is a core field in modern condensed matter physics. Free fermion systems, such as topological insulators and superconductors, have been studied using the "Tenfold Way" and K-theory. Building on Kitaev's idea of $\Omega$-spectrum and classifying space, as well as Freed-Moore's K-theory, this work demonstrates that free fermionic systems form a genuine $G$-$\Omega$-spectrum and clarifies its connection to several distinct classification schemes appearing in the physical literature. By introducing the $\mathbb{Z}_2$-graded algebra $A_{\mathrm{sym}}^V$, the classification problem for systems with general symmetries, including antilinear symmetries, antisymmetries, projective representations, and point group symmetries, is turned into an extension problem in representation theory. To solve this, a computational method for the $\mathbb{Z}_2$-graded Wedderburn-Artin decomposition of $A_{\mathrm{sym}}^V$ is developed. This decomposition not only yields a classification but also enables the explicit construction of the corresponding Dirac Hamiltonian. Furthermore, a GAP programming package has been developed to automate these calculations.

Swarmalators are a class of coupled oscillators that simultaneously synchronize in both space and phase, providing a minimal model for systems ranging from biological microswimmers to robotic swarms. Time delay is ubiquitous in such systems, arising from finite signal propagation speeds and sensory processing lags, yet its structural form, whether symmetric or asymmetric, has received little attention. Here, we study a one-dimensional swarmalator model with asymmetric time delay, in which the delay enters only the self-interaction terms of the spatial and phase dynamics, breaking the symmetry assumed in prior work. We identify various collective states such as async, static phase wave, static {\pi}, and active {\pi}, and derive analytical stability boundaries for each as a function of the coupling parameters and delay. Our analysis reveals that the asymmetric delay structure fundamentally reshapes the collective phase diagram: in particular, for the asymmetric delay models, increasing the delay systematically expands the active {\pi} state at the expense of other ordered states, in contrast to the symmetric delay model, which more strongly promotes the presence of unsteady states that are generally not well ordered. By providing closed-form stability conditions validated against numerical simulations, our work establishes that the internal structure of the delay, not merely its magnitude, is a decisive factor in determining the emergent collective behavior of swarmalator populations.

In this paper, we focus on one-dimensional vertical infiltration, assuming constant diffusivity and a quadratic relationship between hydraulic conductivity and water content. Under these assumptions, Richards' equation reduces to Burgers' equation, which we then linearize via the Hopf-Cole transformation. This turns the initial boundary value problem into a diffusion equation on a finite interval with mixed boundary conditions. To solve it, we use the Unified Transform Method (also known as the Fokas method). This approach gives an explicit integral representation of the solution, and when evaluated numerically, the results match classical Fourier series solutions exactly, but with better convergence and stability. Two examples from hydrological applications are examined.

Billiards tables - a minimal model for particles moving in a confined region - are known to present classical (and quantum) different features according to their shape, ranging from strongly chaotic to integrable dynamics. Here we consider the role of a stochastic perturbation of the elastic reflection law, and show that while chaotic billiards maintain their key statistical feature, the behaviour for integrable billiard tables is completely different: it can be linked, for tiny perturbations, to Evans stochastic billiard, where at each collision the reflected angle is a uniformly distributed stochastic variable on $(-\pi/2,\pi/2$). The resulting spatial stationary measure has peculiar aspects, like being typically non uniform along the boundary, differently from any chaotic billiard table.

Dynamical systems techniques are a powerful tool to analyse systems of ordinary differential equations, written in an appropriate form. For a given theory of gravity, the cosmological field equations typically lead to a system of ordinary differential equations. Casting these cosmological equations into the form of a dynamical system requires a careful choice of the dynamical variables. Despite this being a critical step, relatively little is said about this process in the literature. We discuss how different variable choices affect the information that can be extracted from the Friedmann equations. We begin by reviewing the standard cosmological model with dark matter, radiation, and dark energy, and include quintessence models. We revisit well-known models with an exponential potential using new variables. This discussion is then extended to models with scalar fields and more intricate coupling terms.

We study the dynamics of the Bianchi~IX (Mixmaster) universe under classical polymerization and generalized uncertainty principle (GUP) deformation of the Poisson brackets. Starting from the Misner Hamiltonian, we derive the effective equations of motion with both modifications and analyze the duration of Kasner epochs as a probe of dynamical behavior. Our results show that GUP corrections typically shorten the epochs, leading to more frequent wall collisions, whereas polymer corrections prolong them and suppress successive bounces. At leading order, the combined deformation produces an additive shift that interpolates between these two trends. While the billiard picture remains robust, the strength of Mixmaster chaos becomes sensitive to the deformation parameters. These results illustrate how Planck-scale corrections may either enhance or suppress cosmological chaos, offering a controlled framework for exploring early-universe dynamics.

Dyadic Green's function is an important tool of computational photonics, giving deeper insights into light-matter interaction. We present an operator approach to the derivation of the dyadic Green's function of a generic anisotropic planarly-layered medium for both electric and magnetic fields. The resulting Green's function is expressed through the evolution operators (a kind of transfer matrices) of the comprising layers and the surface impedance tensors, the singular term being naturally separated from other terms. The operator approach to the Green's function simplifies both the conceptual understanding of the problem and the subsequent practical applications, some of which are demonstrated here. The proposed approach can be easily generalized to the case of spherical and cylindrical layers. The obtained results can be applied in nanophotonics engineering problems.

We discuss various forms of refinements of Vogel's universality in Chern-Simons theory. While the original universality applies to arbitrary simple Lie groups, its counterpart in refined Chyrn-Simons theory is restricted to simply laced Lie groups.

Thermal properties of a two-layered composite conductor are modified in case the interface is damaged. The present paper deals with nondestructive evaluation of perturbations of interface thermal conductance due to the presence of defects. The specimen is heated by means of a lamp system or a laser while its surface temperature is measured with an infrared camera in the typical framework of Active Thermography. Defects affecting the interface are evaluated using an inversion formula obtained by means of Laplace transformation and suitable symmetries of parabolic differential operators (reciprocity). Results of numerical inversion from simulated data are encouraging

This paper provides a systematic investigation of the mathematical structure of path measures and their profound connections to stochastic differential equations (SDEs) through the framework of second-order Hamilton--Jacobi (HJ) equations. This approach establishes a unified methodology for analyzing large deviation principles (LDPs), entropy minimization, and entropy production in stochastic systems. Second-order HJ equations are shown to play a central role in bridging stochastic dynamics and measure theory while forming the foundation of stochastic geometric mechanics and their applications in stochastic thermodynamics. The large deviation rate function is rigorously derived from the probabilistic structure of path measures and proved to be equivalent to the Onsager--Machlup functional of stochastic gradient systems coupled with second-order HJ equations. We revisit entropy minimization problems, including finite time horizon problems and Schrödinger's problem, demonstrating the connections with stochastic geometric mechanics. Furthermore, we present a novel decomposition of entropy production for stochastic systems, revealing that thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations. Together, this work establishes a comprehensive mathematical study of the relations between path measures and stochastic dynamical systems, and their diverse applications in stochastic thermodynamics and beyond.

We consider the Hankel determinant and orthogonal polynomials with respect to the deformed Laguerre weight $w(x; t) = {x^\alpha }{\mathrm e^{ - x}}{(x + t)^\lambda },\; x\in \mathbb{R}^{+} $ with parameters $\alpha > -1,\; t > 0$ and $\lambda \in \mathbb{R}$. This problem originates from the information theory of single-user multiple-input multiple-output (MIMO) systems studied by Chen and McKay [{\em IEEE Trans. Inf. Theory} {\bf 58} ({2012}) {4594--4634}]. By using the ladder operators for orthogonal polynomials with general Laguerre-type weights, we obtain a system of difference equations and a system of differential-difference equations for the recurrence coefficients $\alpha_n(t)$ and $\beta_n(t)$. We also show that the orthogonal polynomials satisfy a second-order ordinary differential equation. By using Dyson's Coulomb fluid approach, we obtain the large $n$ asymptotic expansions of the recurrence coefficients $\alpha_n(t)$ and $\beta_n(t)$, the sub-leading coefficient $\mathrm p(n, t)$ of the monic orthogonal polynomials, the Hankel determinant $D_n(t)$ and the normalized constant $h_n(t)$ for fixed $t\in\mathbb{R}^{+}$. We also discuss the long-time asymptotics of these quantities as $t\rightarrow\infty$ for fixed $n\in\mathbb{N}$. The large $n$ and large $t$ asymptotics of the above quantities are very important for the study of the asymptotics of the mutual information distribution and two fundamental quantities (the outage capacity and the error probability) for single-user MIMO systems.

Brzeziński's trusses are ``ring-like'' algebraic structures in which the addition is replaced with an abelian heap operation and the binary product satisfies a natural distributivity rule of the ternary product. The question of how to define ($\mathbb{Z}_2$-graded) super-versions of trusses is addressed in this note. Taking our cue from the theory of algebraic supergroups, we define an affine supertruss as a representable functor from the category of unital associative supercommutative superalgebras to the category of trusses. The representing superalgebras are equipped with a `cotruss' structure--a new concept in itself. We show that from an affine supertruss one can construct an affine superbrace, and so generalise Rump's braces to supermathematics. As an application of these constructions, we propose a generalisation of the set-theoretic Yang--Baxter equation to the setting of affine superschemes.

We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators $H=(i\partial+A(x))^2+V(x)$ on $\mathbb{R}^{n}$, $n\ge3$. Under suitable decay assumptions on the electric and magnetic potentials, and excluding a threshold resonance at zero, we show that for all $z \in \mathbb{C}\setminus[0,+\infty)$, \begin{equation*} \|(H-z)^{-1}\phi\|_{L^{q}}\lesssim|z|^{\theta(p,q)} (1+|z|^{\gamma}) \|\phi\|_{L^{p}} \end{equation*} throughout the full free resolvent range $(\frac1p,\frac1q)\in\Delta(n)$, where $\theta(p,q)=\frac n2(\frac1p-\frac1q)-1$. Here $\gamma=\frac 12\frac{n-1}{n+1}$ under the basic magnetic decay hypothesis, or $\gamma=\frac{n-1}{4n}$ under a different decay assumption on $A(x)$; for the second case we use a weak endpoint estimate of Frank--Simon type \begin{equation*} \|R_{0}(z)\phi\| _{L^{\frac{2n}{n-1},\infty}_{r}L^{2}_{\omega}} \lesssim |z|^{-\frac12} \|\phi\|_{L^{\frac{2n}{n+1},1}_{r}L^{2}_{\omega}}. \end{equation*} The result extends the known electromagnetic estimates from fixed frequency and a smaller exponent region to all frequencies and the full Kenig--Ruiz--Sogge range. We also prove a variant with weaker local assumptions in a smaller range $\Delta_1(n)$. As applications, we obtain $L^p-L^{p'}$ restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations.

We present an elementary approach to observe frequency cascade on forced nonlinear Schr{ö}dinger equations. The forcing term (which may also appear as a potential term instead) consists of a constant term, perturbed by a modulated Gaussian well. Algebraic computations provide an explicit frequency cascade when time and space derivatives are discarded from the nonlinear Schr{ö}dinger equation. We provide stability results, showing that when derivatives are incorporated in the model, the initial algebraic solution may be little affected, possibly over long time intervals. Numerical simulations are provided, which support the analysis.

Iterated Graph Systems (IGS) transplant ideas from fractal geometry into graph theory. Building on this framework, we extend Edge IGS from the primitive to the reducible setting. Within this broader context, we formulate rigorous definitions of multifractality and multiscale-freeness for fractal graphs, and we establish conditions that are equivalent to the occurrence of these two phenomena. We further determine the corresponding fractal and degree spectra, proving that both are finite and discrete. These results complete the foundational theory of Edge IGS by filling the gap left by the primitive case studied in [1, 2].

We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating $F_K (y,z,q)$ to $\hat{Z}(q)$ invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple $Spin^c$ decorated TQFT from the three variable series. We observe that the super $F_K (y,z,q)$ itself and its results exhibit distinctive features compared to the GM series.

The quantum expectation value and the stationary noise spectral density for a Fabry-P'erot gravitational-wave detector with a DC readout scheme are discussed in detail only through the quantum electrodynamics of lasers and the Heisenberg equations of mirrors' motion. We demonstrate that the initial conditions of the mirrors' motion concentrate around the fundamental frequency of the pendulum and are not related to the frequency range of our interest. Although, in the ideal case, there is consensus that the shot-noise contribution from the laser to the high-frequency range of the signal-referred noise spectral density decreases as the injected laser power increases, our derived noise spectral density shows that the shot-noise contribution does not decrease. This is due to leakage of classical radiation pressure forces from the carrier field to the output port, and the carrier field is used as the reference in the DC readout scheme. Since classical radiation pressure acts as a constant force, it shifts the pendulum's equilibrium point of the mirrors' motion. To recover the ideal case, we must consider adjusting the interferometer's tuning point to place the mirrors at their equilibrium positions. We investigate the case where the equilibrium tuning is incomplete and show that the behavior of the above shot noise is due to this incompleteness. We also discuss the maximum deviation of the mirror displacements from the equilibrium point during incomplete tuning to recover a near-ideal case.

In this article, we put forward a practical but generic approach towards constructing a large family of $(3+1)$ dimension lattice models which can naturally lead to a single Weyl cone in the infrared (IR) limit. Our proposal relies on spontaneous charge $U(1)$ symmetry breaking to evade the usual no-go theorem of a single Weyl cone in a 3d lattice. We have explored three concrete paths in this approach, all involving fermionic topological symmetry protected states (SPTs). Path a) is to push a gapped SPT in a 3d lattice with time-reversal symmetry (or $T$-symmetry) to a gapless topological quantum critical point (tQCP) which involves a minimum change of topologies,i.e. $\delta N_w=2$ where $\delta N_w$ is the change of winding numbers across the tQCP. Path b) is to peal off excessive degrees of freedom in the gapped SPT via applying $T$-symmetry breaking fields which naturally result in a pair of gapless nodal points of real fermions. Path c) is a hybrid of a) and b) where tQCPs, with $\delta N_w \geq 2$, are further subject to time-reversal-symmetry breaking actions. In the infrared limit, all the lattice models with single Weyl fermions studied here are isomorphic to either a tQCP in a DIII class topological superconductor with a protecting $T$-symmetry, or its dual, a $T$-symmetry breaking superconducting nodal point phase, and therefore form an equivalent class. For a generic $T$-symmetric tQCP along Path a), the conserved-charge operators span a six-dimensional linear space while for a $T$-symmetry breaking gapless state along Path b), c), charge operators typically span a two-dimensional linear space instead. Finally, we pinpoint connections between three spatial dimensional lattice chiral fermion models and gapless real fermions that can naturally appear in superfluids or superconductors studied previously.

We study the high-frequency Fourier asymptotics of imaginary Gaussian multiplicative chaos on the unit circle, a complex-valued random distribution formally given by $\mathrm M_{\mathrm i\beta}=\exp(\mathrm i\beta X)$, where $X$ is a log-correlated Gaussian field. In the subcritical phase $\beta\in(0,1)$, we prove that its Fourier dimension, defined by the optimal polynomial decay exponent of $|\widehat{\mathrm M_{\mathrm i\beta}}(n)|^2$, is almost surely equal to $1-\beta^2$. This result holds for a broad class of log-correlated fields whose covariance differs from the exact logarithmic kernel by a sufficiently regular function. For the exactly log-correlated field on the circle, we obtain the following results. We prove that the chaos almost surely fails to belong to $H^{-\beta^2/2}(\mathbb T)$, the critical Sobolev space left open by previous regularity results. We further establish a central limit theorem: the rescaled coefficients $n^{(1-\beta^2)/2}\widehat{\mathrm M_{\mathrm i\beta}}(n)$ converge in law to an isotropic complex Gaussian random variable, and finitely many consecutive coefficients converge jointly to independent copies. The high-frequency content of $\mathrm M_{\mathrm i\beta}$ behaves as a white noise: $n^{(1-\beta^2)/2}e^{\mathrm ii n\theta}\mathrm M_{\mathrm i\beta}$ converges in $H^s(\mathbb T)$, $s<-1/2$, to a complex white noise with explicit intensity $\kappa(\beta)=\frac{1}{\pi}\Gamma(1-\beta^2)\sin\big(\frac{\pi\beta^2}{2}\big)$. The proof relies on moment identities obtained from Coulomb-gas integrals and Jack-polynomial expansions. Their asymptotic analysis is governed by partitions with large gaps, where the Pieri coefficients appearing in these expansions simplify, and the leading contribution becomes explicit.

We investigate proper infall times in the Schwarzschild and Kerr spacetimes from a covariant perspective, focusing on the role of black--hole rotation in the focusing properties of timelike geodesic this http URL perform a geometrically consistent comparison between rotating and non--rotating black holes, we analyse infall trajectories between surfaces of equal circumferential radius in the equatorial plane. Using equatorial timelike geodesics in the test--particle limit, we compute and compare the corresponding proper infall times for different values of the specific energy $E$, specific angular momentum $L$, and black--hole spin parameter $a$. Within the equal circumferential-radius prescription adopted here, we show that Kerr angular momentum $a$ can produce longer or shorter integrated proper infall times relative to the Schwarzschild case, depending on the orbital configuration and energy regime considered. We then interpret these results within the covariant $1+3$ formalism of general relativity, in terms of the expansion, shear, and Raychaudhuri evolution of timelike congruences. Our analysis shows that the Kerr--Schwarzschild differences in proper infall times are encoded in the corresponding Raychaudhuri time integrand, which reflects a competition between the radial evolution of the expansion and the nonlinear focusing contribution driven by expansion and shear. Black--hole rotation modifies both effects in a systematic way, leading to distinct behaviours for prograde and retrograde infall configurations.

We construct a parametrix of an elliptic Gevrey pseudodifferential operator, by introducing a family of norms for formal Gevrey symbols with the property of a Banach algebra under the symbol calculus. As an application, we obtain exponential estimates for adiabatic projectors in the Gevrey setting.

We derive the bosonic sector of the AdS$_3$/CFT$_2$ correspondence from the $(1+1)$-dimensional Gross-Neveu (GN) model with $N$ fermion species and a local quartic interaction, with no stringy or geometric input. A Bargmann-Wigner fusion scheme generates an infinite tower of higher-spin composite fields with a linear Regge trajectory. Competition between spin-0 (chiral) condensation and spin-1 pairing defines an emergent radial coordinate; local fluctuations of this condensate ratio, tracked by a comoving derivative, generate the AdS$_3$ line element. The large-$N$ species sum promotes $z$ from a parameter to a genuine bulk dimension. We show that the full $SO(2,2)$ bulk isometry group, whose special conformal generators mix $z$ with the boundary GN coordinates, emerges from local symmetries of the boundary condensates, and holographic RG flow identifies $z$ with the Wilsonian cutoff this http URL find that a hierarchy of phase transitions in the enlarged GN model map to a bulk description: spin-2 decoherence $\to$ spin-1 decoherence $\to$ chiral symmetry restoration occur at the Hawking-Page, Hagedorn, and Planck temperatures in the bulk picture, respectively, represented as a layered radial profile of the bulk geometry, with successive condensate sectors dissolving at progressively greater depths into the bulk.

We study non-Hermitian quantum mechanics of an inverted triple-well potential within the exact WKB framework. For a single classical potential, different Siegert boundary conditions define three distinct quantum problems: the PT-symmetric, resonance, and anti-resonance systems. For each case, we derive the exact quantization condition and construct the associated trans-series solution. By identifying the resurgent structures and cancellations in these non-Hermitian setups, we obtain the median-summed series, clarifying when the spectra are real or complex in accordance with the physical properties of each system. Establishing explicit links to the semi-classical path integral formalism, we elucidate the roles of bounce and bion configurations in these non-Hermitian systems. This analysis predicts PT-symmetry breaking, which we also verify numerically. Using the median quantization conditions, we prove the existence of this symmetry breaking and establish an exact equation for the exceptional point, which emerges as a remarkably simple algebraic relation between the bounce and bion actions. We further show that the median-summed non-perturbative correction to the spectrum vanishes at the exceptional point, while the resurgent structure survives through a universal minimal trans-series. For the resonance and anti-resonance systems, we find that the exact median-summed spectra are related by complex conjugation, representing time reversal in this setting, are necessarily complex, and do not exhibit an exceptional point. Although their spectra differ significantly from the PT-symmetric case, they share the same minimal trans-series. By maintaining explicit links with the path integral saddles and the formal theory of resurgence, our analysis provides a unified and general perspective on the quantization of non-Hermitian theories.

Many complex systems are described by Langevin-type equations in which the noise exhibits long-range correlations and couples to the system in a state-dependent, multiplicative manner, leading to heterogeneous non-Markovian diffusion. Here, we investigate the problem of diffusion driven by fractional Gaussian noise with a general multiplicative coefficient from a path-integral perspective. Using a stationary-phase approximation, we derive a Gaussian propagator expressed in terms of the Lamperti transform of the process. In the additive limit, our results recover the path-integral representation of fractional Brownian motion based on its Riemann-Liouville formulation and establish its equivalence with the Langevin construction. We further analyze the effect of subordinating the process to a killing rate within the Feynman-Kac framework, and develop a general procedure to derive kinetic equations in terms of effective local Hamiltonians. We show that the interplay between multiplicative diffusion and confinement induces an effective drift term, leading to probability accumulation in regions of low noise amplitude.

Single-shot quantum information theory is governed not only by entropy exponents, but also by the finite-resource constants that multiply them. These constants directly affect the quantitative performance of decoupling, covering, convex-splitting, position-based decoding, and one-shot communication protocols, yet they are often inherited from nonoptimal scalar estimates or from classical-to-quantum lifting arguments that introduce additional losses. In this work we show that the operator layer-cake representation provides a mechanism for lifting sharp scalar inequalities to the noncommutative setting without loss. Using an iterative Riemann--Stieltjes integration-by-parts method, we derive sharp quantum trace inequalities that tighten several standard single-shot bounds. For a logarithmic trace inequality recently introduced by Cheng \emph{et al.}\ and subsequently used in quantum covering and decoupling problems, we determine the exact optimal prefactor, replacing the previously known constant by a smaller Lambert-$W$ constant and proving universal optimality for positive operators. We also completely characterize the threshold behavior that appears under normalization to quantum states. In addition, we establish optimal two-sided collision-divergence inequalities, which lead to improved position-based decoding and single-shot classical communication bounds. These results show that several finite-resource bounds in single-shot quantum information can be tightened, and that within the layer-cake Rényi-divergence framework the resulting constants are genuine optimality barriers rather than artifacts of the proof.