We present an explicit, fully local Reeh-Schlieder approximation scheme for coherent states of a free scalar field. For any bounded region $\mathcal{U}$, we construct a one-parameter family of bounded operators $\hat{A}_\zeta$ localized in the causal complement of $\mathcal{U}$. The action of $\hat{A}_\zeta$ on the vacuum approximates the target coherent state in the limit $\zeta \to 0$.

In this work, we characterize self-adjoint operators that commute with magnetic translations. We use this characterization to derive effective kinetic energy functionals for homogeneous electron gases and three-dimensional electronic systems with two-dimensional symmetries in the presence of a magnetic field.

Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley's theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation-based distances compared to the conventional generator-based distances in finite groups. The reason why we are interested in distances on groups is that finite groups appear in symbolic representations of time series, most notably in the so-called ordinal representations, whose symbols are precisely permutations, usually called ordinal patterns in that context. The natural extension from groups to group-valued time series is also discussed, as well as how such metric tools can be applied in time series analysis. Both theory and applications are illustrated with examples and numerical simulations.

The open XXZ spin chain with the anisotropy parameter $\Delta=-\frac12$, diagonal boundary fields that depend on a parameter $x$, and finite length $N$ is studied. In a natural normalisation, the components of its ground-state vector are polynomials in $x$ with integer coefficients. It is shown that their sum is given by a generating function for the weighted enumeration of totally-symmetric alternating sign matrices with weights depending on $x$.

We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature $\beta>2$. In that regime, the partition function diverges and the interaction needs to be cut off at a length scale $\lambda>0$. Particles of opposite charges tend to pair into dipoles of length scale comparable to $\lambda$, which themselves can aggregate into multipoles. Despite the slow decay of dipole--dipole interactions, we construct a convergent cluster expansion around a hierarchical reference model that retains only intra-multipole interactions. This yields a large deviation result for the number of $2p$-poles as well as a sharp free energy expansion as $N\to\infty$ and $\lambda\to0$ with three contributions: (i) the free energy of $N$ independent dipoles, (ii) a perturbative correction, and (iii) the contribution of a non-dilute subsystem. The perturbative term has two equivalent characterizations: (a) a convergent Mayer series obtained by expanding around an i.i.d.\ dipole model; and (b) a variational formula as the minimum of a large-deviation rate function for the empirical counts of $2p$-poles. The Mayer coefficients exhibit transitions at $\beta_p=4-\tfrac{2}{p}$, that accumulate at $\beta=4$, which corresponds to the Berezinskii-Kosterlitz-Thouless transition in the low-dipole-density limit. At $\beta=\beta_p$ the $p$-dipole cluster integrals switch from non-integrable to integrable tails. The non-dilute system corresponds to the contribution of large dipoles: we exhibit a new critical length scale $R_{\beta, \lambda}$ which transitions from $\lambda^{-(\beta-2)/(4-\beta)}$ to $+\infty$ as $\beta$ crosses the critical inverse temperature $\beta=4$, and which can be interpreted as the maximal scale such that the dipoles of that scale form a dilute set.

We study the asymptotic behavior of continuous-time, time-inhomogeneous Markovian quantum dynamics in a stationary random environment. Under mild faithfulness and eventually positivity-improving assumptions, the normalized evolution converges almost surely to a stationary family of full-rank states, and the normalized propagators converge almost surely to a rank-one family determined by these states. Beyond a disorder-dependent threshold, these convergences occur at exponential rates that may depend on the disorder; when the environment is ergodic, the rate itself is deterministic. When the dynamical propagators display vanishing maximal temporal stochastic correlation, convergence in stochastic expectations for the above limits is faster than any power of the time separation, and improves to exponential rates when the dynamical propagators display stochastically independent increments. These expectation bounds yield disorder-uniform high-probability estimates. The framework does not require complete positivity or trace preservation and encompasses random Lindbladian evolutions and collision-model dynamics.

We study FK-percolation where the edge parameters are chosen as independent random variables in the near-critical regime. We show that if these parameters satisfy a natural centering condition around the critical point, then the quenched model typically exhibits critical behaviour at scales much larger than the deterministic characteristic length. More precisely, in a box of size $N$, if the homogeneous model with deterministic edge parameter $p$ looks critical in the regime $|p-p_c|\le \textrm W$, then the quenched model with random edge parameters $\mathbf p$ that typically satisfy $|\mathbf p-p_c|\le \textrm W^{1/3}$ looks critical, assuming some conjectured inequality on critical exponents, and up to logarithmic corrections. We also treat the special case of Bernoulli percolation, where we show that if one first samples non-degenerate independent random edge parameters centered around $\frac12$, and then a percolation configuration on these edges, the quenched model almost surely looks critical at large scales.

We study the q-characters and modular data of exceptional W-algebras and give several examples and applications. We establish equality of q-characters and modular data between certain boundary W-algebras, leading in particular to a largely complete determination of fusion rules of exceptional W-algebras in type A.

We consider Gamow's liquid drop functional, $\mathcal{E}$, on $\mathbb{R}^3$ and construct non-minimizing, volume constrained, critical points for volumes $3.512 \cong \alpha_0 < V < 10$. In this range, we establish a mountain pass set up between a ball of volume $V$ and two balls of volume $V/2$ infinitely far apart. Intuitively, our critical point corresponds to the maximal energy configuration of an atom of volume $V$ as it undergoes fission into two atoms of volume $V/2$. Our proof relies on geometric measure theoretical methods from the min-max construction of minimal surfaces, and along the way, we address issues of non-compactness, ``pull tight" with a volume constraint, and multiplicity.

We provide an index-theoretic proof of the bulk-boundary correspondence for two- and three-dimensional second-order topological insulators that preserve inversion symmetry, which are modeled as rectangles and rectangular prism-shaped systems. Our method uses extensions of the symbols of some Toeplitz operators on discrete quarter planes and computations of topological equivariant K-theory groups.

This work undertakes a twofold investigation. In the first part, we examine the inequalities and uncertainty principles in the framework of offset linear canonical transform (OLCT), with particular attention to its scaling and shifting effects. Theoretical developments are complemented by numerical simulations that substantiate and illustrate the analytical results. In the second part, we establish the connection of quaternion offset linear canonical transform (QOLCT) and the OLCT by employing the orthogonal plane split (OPS) approach. Through this approach, the inequalities and uncertainty principles derived for the OLCT are extended to the QOLCT. Moreover, the computational methods designed for the OLCT may be systematically adapted to facilitate the numerical implementation of the QOLCT using this connection between OLCT and QOLCT.

We show that the Mirzakhani volume, as introduced by Chekhov, of the moduli space of every non-contractible crowned hyperbolic surface is naturally expressible as a sum of Gaussian rational multiples of polylogarithms evaluated at $\pm1$ and $\pm\sqrt{-1}$.

We extend the results of Lagrangian formulations study to construct gauge-invariant Lagrangians for (ir)reducible integer higher-spin massless and massive representations of the Poincare group with a Young tableau $Y[\hat{s}_1,\hat{s}_2,\hat{s}_3]$ in $d$-dimensional flat space-time (as the probable candidates to describe the Dark Matter problem beyond the SM). These particles are described within a metric-like formulation by tensor fields with 3 groups of antisymmetric Lorentz indices $\Phi_{\mu^1[{\hat{s}_1}],\mu^2[{\hat{s}_2}], \mu^3[{\hat{s}_3}]}$ on a basis of the BRST method with complete, $Q$, and incomplete, $Q_c$, BRST operators. We found unconstrained (with $Q$) and constrained (with $Q_c$ and off-shell BRST invariant holonomic constraints) gauge Lagrangian formulations with different configuration spaces and reducibility stages. The deformation procedure to construct interacting gauge model with mixed-antisymmetric fields is proposed.

This is the more technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrodinger equation. Combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. The companion paper gives applications. Our main technical tools are three new pseudodifferential calculi, $\Psi_{\natural}$ (a variant of the semiclassical scattering calculus), $\Psi_{\natural\mathrm{res}}$, and $\Psi_{\natural2\mathrm{res}}$, the latter two of which are created by ``second microlocalizing'' the first at certain locations. This paper and the companion paper can be read in either order, since the latter treats the former as a black box.

We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems.

We investigate Snyder space-time and its generalizations, including Yang and Snyder-de-Sitter spaces, which constitute manifestly Lorenz invariant noncommutative geometries. This work initiates a systematic study of gauge theory on such spaces in the semi-classical regime, formulated as Poisson gauge theory. As a first step, we construct the symplectic realization of the relevant noncommutative spaces, a prerequisite for defining Poisson gauge transformations and field strengths. We present a general method for representing the Snyder algebra and its extensions in terms of canonical phase space variables, enabling both the reproduction of known representations and the derivation of novel ones. These canonical constructions are employed to obtain explicit symplectic realizations for the Snyder-de-Sitter space and construct the deformed partial derivative which differentiates the underlying Poisson structure. Furthermore, we analyze the motion of freely falling particles in these backgrounds and comment on the geometry of the associated spaces.

Consider balls $\Lambda_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and place edges independently between each pair of vertices $x\neq y\in\Lambda_n$ with probability $1-\exp(-\beta J(x, y) )$ where $J(x, y) \asymp \| x-y \|^{-\alpha}$ for some $0<\alpha<5d/6$. We identify the critical window for this model and establish the metric scaling limit within the critical window. More precisely, we show that the metric scaling limit of the maximal components is Brownian, and that this model belongs to the Erdos-Renyi universality class when $0<\alpha<5d/6$. It was recently conjectured by Hutchcroft [43, Section~7.1] that this model is a member of the Erdos-Renyi universality class when $\alpha\in(d, 4d/3]$, and we believe that this is also true for all $\alpha\in (0, d]$. This paper proves a first result in that direction. We further show that when $0<\alpha<2d/3$, the girth of each maximal component in the critical window is $\Omega_P(|\Lambda_n|^{1/3})$, contrary to the situation when $d<\alpha\leq 4d/3$ where the girth equals $3$ . Our key tool is a universality principle proven in [10, 13]. Our proof technique can be adapted for the analogous problem on long-range percolation on the discrete torus provided an upper bound on the barely subcritical two-point function is obtained.

We show that, for sudden quenches, the work distribution reduces to the statistics of traces of powers of Haar unitaries, which are random unitary matrices drawn uniformly from the unitary group. For translation-invariant quadratic fermionic chains with interactions extending to $m$ neighbors and periodic boundary conditions, the Loschmidt amplitude admits a unitary matrix-model / Toeplitz representation, which yields a work variable of the form $W=\sum_{r\le m} a_r\,\mathrm{Re}\,\mathrm{Tr}\,U^r$ (and in models with pairing terms -- superconducting pairing -- additional $b_r\,\mathrm{Im}\,\mathrm{Tr}\,U^r$ terms appear). By invoking multivariate central limit theorems for vectors of traces of unitaries, we obtain a Gaussian distribution for $P(W)$ with variance $\mathrm{Var}(W)=\frac{1}{2}\sum_r r\,(a_r^2+b_r^2)$ and asymptotic independence across different powers. We also characterise the conditions under which non-Gaussian tails arise, for example from many interaction terms or their slow decay, as well as the appearance of Fisher--Hartwig singularities. We illustrate these mechanisms in the XY chain. Various numerical diagnostics support the analytical results.

Mixed states can exhibit two distinct kinds of symmetries, either on the level of the individual states (strong symmetry), or only on the level of the ensemble (weak symmetry). Strong symmetries can be spontaneously broken down to weak ones, a mechanism referred to as Strong-to-Weak Spontaneous Symmetry Breaking (SW-SSB). In this work, we first show that maximally mixed symmetric density matrices, which appear, for example, as steady states of symmetric random quantum circuits have SW-SSB when the symmetry is an on-site representation of a compact Lie or finite group. We then show that this can be regarded as an isolated point within an entire SW-SSB phase that is stable to more general quantum operations such as measurements followed by weak postselection. With sufficiently strong postselection, a second-order transition can be driven to a phase where the steady state is strongly symmetric. We provide analytical and numerical results for such SW-SSB phases and their transitions for both abelian $\mathbb{Z}_2$ and non-abelian $S_3$ symmetries in the steady state of Brownian random quantum circuits with measurements. We also show that such continuous SW-SSB transitions are absent in the steady-state of general strongly symmetric, trace-preserving quantum channels (including unital, Brownian, or Lindbladian dynamics) by analyzing the degeneracies of the steady states in the presence of symmetries. Our results demonstrate robust SW-SSB phases and their transitions in the steady states of noisy quantum operations, and provide a framework for realizing various kinds of mixed-state quantum phases based on their symmetries.

We present a constructive method to maximize the expectation value of operators that implement a symmetry on a subsystem, making use of modular tools. More generally, we study the positive cones associated with a von Neumann algebra, as defined by Araki. Given a reference vector, an algebra, and a state on the algebra, the purification of the state in the cone $\alpha = 0$, associated with the reference vector and the algebra, yields the unique vector whose overlap with the reference vector is maximal among all possible purifications. This establishes that the supremum in Uhlmann's theorem is uniquely attained by this vector, thereby providing the fidelity between the given state and the state obtained by restricting the reference vector to the algebra. Moreover, this purification can be explicitly constructed using modular tools. In addition, given an automorphism of the algebra, we show how to construct isometries implementing the automorphism using the positive cones. We prove that the isometry constructed from the cone $\alpha = 0$ is the one with maximal expectation value among all possible isometries implementing the automorphism. We illustrate these ideas with two simple examples: one involving a system of two spins, and the other in the theory of the massless scalar field in 3+1 dimensions.

A variety of physical phenomena, such as amplification, absorption, and radiation, can be effectively described using non-Hermitian operators. However, the introduction of non-uniform non-Hermiticity can lead to the formation of exceptional points in a system's spectrum, where two or more eigenvalues become degenerate and their associated eigenvectors coallesce causing the underlying operator or matrix to become defective. Here, we explore extensions of the Clifford and quadratic $\epsilon$-pseudospectrum, previously defined for Hermitian operators, to accommodate non-Hermitian operators and matrices, including the possibility that the underlying operators may possess exceptional points in their spectra. In particular, we provide a framework for finding approximate joint eigenstates of a $d$-tuple of Hermitian operators $\mathbf{A}$ and non-Hermitian operators $\mathbf{B}$, and show that their Clifford and quadratic $\epsilon$-pseudospectra are still well-defined despite any non-normality. We prove that the non-Hermitian quadratic gap is local with respect to the probe location when there are perturbations to one or more of the underlying operators. Altogether, this framework enables the exploration of non-Hermitian physical systems' $\epsilon$-pseudospectra, including but not limited to photonic systems where gain, loss, and radiation are prominent physical phenomena.

We study invariant random matrix ensembles \begin{equation*} \mathbb{P}_n(d M)=Z_n^{-1}\exp(-n\,tr(V(M)))\,d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the underlying density of states is one-cut regular. Considering the average \begin{equation*} E_n[\phi;\lambda,\alpha,\beta]:=\mathbb{E}_n\bigg(\prod_{\ell=1}^n\big(1-\phi(\lambda_{\ell}(M))\big)\omega_{\alpha\beta}(\lambda_{\ell}(M)-\lambda)\bigg),\ \ \ \ \ \omega_{\alpha\beta}(x):=|x|^{\alpha}\begin{cases}1,&x<0\\ \beta,&x\geq 0\end{cases}, \end{equation*} taken with respect to the above law and where $\phi$ is a suitable test function, we evaluate its large-$n$ asymptotic assuming that $\lambda$ lies within the soft edge boundary layer, and $(\alpha,\beta)\in\mathbb{R}\times\mathbb{C}$ satisfy $\alpha>-1,\beta\notin(-\infty,0)$. Our results are obtained by using Riemann-Hilbert problems for orthogonal polynomials and integrable operators and they extend previous results of Forrester and Witte \cite{FW} that were obtained by an application of Okamoto's $\tau$-function theory. A key role throughout is played by distinguished solutions to the Painlevé-XXXIV equation.

We study how the fusion 2-category symmetry of a fermionic (2+1)d QFT can be affected when one allows for stacking with TQFTs to be an equivalence relation for QFTs. Focusing on a simple kind of fermionic fusion 2-category described purely by group theoretical data, our results reveal that by allowing for stacking with $\mathrm{Spin}(n)_1$ as an equivalence relation enables a finite set of inequivalent modifications to the original fusion 2-categorical-symmetry. To put our results in a broader context, we relate the order of the symmetry modifications to the image of a map between groups of minimal nondegenerate extensions, and to the tangential structure set by the initial categorical symmetry on the background manifold for the QFT.

The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mariño and Vafa as a means to calculate open Gromov-Witten invariants of toric Calabi-Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov-Witten invariants of log Calabi-Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov-Witten invariants.

Spin glass models with quadratic-type Hamiltonians are disordered statistical physics systems with competing ferromagnetic and anti-ferromagnetic spin interactions. The corresponding Gibbs measures belong to the exponential family parametrized by (inverse) temperature $\beta>0$ and external field $h\in\mathbb{R}$. Given a sample from these Gibbs measures, a statistically fundamental question is to infer the temperature and external field parameters. In 2007, Chatterjee (Ann. Statist. 35 (2007), no.5, 1931-1946) first proved that in the absence of external field $h=0$, the maximum pseudolikelihood estimator for $\beta$ is $\sqrt{N}$-consistent under some mild assumptions on the disorder matrices. It was left open whether the same method can be used to estimate the temperature and external field simultaneously. In this paper, under some easily verifiable conditions, we prove that the bivariate maximum pseudolikelihood estimator is indeed jointly $\sqrt{N}$-consistent for the temperature and external field parameters. The examples cover the classical Sherrington-Kirkpatrick model and its diluted variants.

In this work, we explore the intriguing phenomenon of acceleration radiation exhibited by an atom falling into a black hole, as previously studied in Phys. Rev. Lett. 121, 071301 (2018) . Our investigation focuses on examining the impact of Lorentz violation within the framework of the bumblebee gravity model on this phenomenon. We observe that the excitation probability although acquires Planck-like factor the exponential part of it acquires the Lorentz violation factor dependent frequency. However, equivalence principle is not violated. Then we calculate the horizon brightened acceleration radiation (HBAR) entropy for this black hole geometry. We observed that the HBAR entropy has the form similar to that of Bekenstein-Hawking black hole entropy however it has been observed that it is also influenced by Lorentz violation associated to the Bumblebee theory. Additionally, we note that the Lorentz violation effect and conformal symmetry both affect the transition probabilities of a two-level atomic detector.

We construct a topological quantum field theory based on the moduli space of $\mathrm{Spin}(7)$-instantons on 8-dimensional manifolds. Using the Mathai-Quillen formalism, we derive the action of the theory in purely geometric terms, which coincides with prior results in the literature. We then reformulate the theory within the AKSZ formalism, obtaining a Batalin-Vilkovisky action that, after gauge fixing, matches our Mathai-Quillen construction while making the BRST symmetry explicit and providing a natural framework for classical observables. We also show that the Batalin-Vilkovisky action can be elegantly recast as a Chern-Simons type theory.

We investigate the propagation of Love waves in an isotropic half-space modelled as a linear {elastic isotropic} Cosserat material. To this aim, we show that a method commonly used to study Rayleigh wave propagation is also applicable to the analysis of Love wave propagation. This approach is based on the explicit solution of an algebraic Riccati equation, which operates independently of the traditional Stroh formalism. The method provides a straightforward numerical algorithm to determine the wave amplitudes and speed{s}. Beyond its numerical simplicity, the method guarantees the existence and uniqueness of a subsonic wave speed, addressing a problem that remains unresolved in most Cosserat solids generalised {continua} theories. Although often overlooked, proving the existence of an admissible solution is, in fact, the key point that validates or invalidates the entire analytical approach used to derive the equation determining the wave speed. Interestingly, it is confirmed that the Love waves do not need the artificial introduction of a surface layer, as indicated in the literature.

In this letter we consider the time dependent Kondo model where a magnetic impurity interacts with the electrons through a time dependent interaction strength $J(t)$. We develop a new framework based on Bethe ansatz and construct an exact solution to the time-dependent Schrodinger equation. We show that when periodic boundary conditions are applied, the consistency of the solution results in a constraint equation which relates the amplitudes corresponding to a certain ordering of the particles in the configuration space. This constraint equation takes the form of a matrix difference equation, and the associated consistency conditions restrict the interaction strength $J(t)$ for the system to be integrable. For a given $J(t)$ satisfying these constraints, the solution to the matrix difference equations provides the exact many-body wavefunction that satisfies the time-dependent Schrodinger equation. We provide a concrete example of $J(t)$ which satisfies these constraint equations. We show that in this case, the matrix difference equations turn into quantum Knizhnik-Zamolodchikov (qKZ) equations, which are well studied in the literature. The framework developed in this work allows one to probe the non-equilibrium physics of the Kondo model, and being general, it also allows one to solve new class of Hamiltonians with time-dependent interaction strength which are based on quantum Yang-Baxter algebra.

Contrary to the established view of the Lorenz system as an archetype of dissipative chaos lacking conserved quantities, this work rigorously demonstrates the existence of a novel class of history-dependent dynamical invariants. Through a constructive method that augments the phase space, we derive a non-local invariant whose value remains constant along any trajectory. Its history-dependence arises from an integral term that accumulates the orbit's past, thereby ensuring its conservation. The invariant's constancy is verified with high-precision numerical simulations for both periodic and chaotic orbits. This finding reveals a hidden structure within the attractor and affords a new physical interpretation where unstable periodic orbits (UPOs) correspond to specific values of this conserved quantity. The result redefines the notion of non-integrability in dissipative systems, showing that non-local order can coexist with chaotic behavior.

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to multi-matrix models case, which corresponds to the multi-component UC hierarchy.

While the exact separability probability of 8/33 for two-qubit states under the Hilbert-Schmidt measure has been reported by Huong and Khoi [\href{this https URL}{J.Phys.A:this http URL.{\bf57}, 445304(2024)}], detailed derivations remain inaccessible for general audiences. This paper provides a comprehensive, self-contained derivation of this result, elucidating the underlying geometric and probabilistic structures. We achieve this by developing a framework centered on the computation of Hilbert-Schmidt volumes for key components: the quantum state space, relevant flag manifolds, and regular (co)adjoint orbits. Crucially, we establish and leverage the connection between these Hilbert-Schmidt volumes and the symplectic volumes of the corresponding regular co-adjoint orbits, formalized through the Duistermaat-Heckman measure. By meticulously synthesizing these volume computations -- specifically, the ratios defining the relevant probability measures -- we reconstruct and rigorously verify the 8/33 separability probability. Our approach offers a transparent pathway to this fundamental constant, detailing the interplay between symplectic geometry, representation theory, and quantum probability.

We introduce the notion of a Young generating function for a probability measure on integer partitions. We use this object to characterize probability distributions over integer partitions satisfying a law of large numbers and those that satisfy a central limit theorem. We further establish a multilevel central limit theorem, which enables the study of random standard Young tableaux. As applications of these results, we describe the fluctuations of height functions associated with (i) the Plancherel growth process, (ii) random standard Young tableaux of fixed shape, and (iii) probability distributions induced by extreme characters of the infinite symmetric group $S_\infty$. In all cases, we identify the limiting fluctuations as a conditioned Gaussian Free Field.

The Koopman operator framework offers a way to represent a nonlinear system as a linear one. The key to this simplification lies in the identification of eigenfunctions. While various data-driven algorithms have been developed for this problem, a theoretical characterization of Koopman eigenfunctions from geometric properties of the flow is still missing. This paper provides such a characterization by establishing an equivalence between a set of Koopman eigenfunctions and a set of commuting symmetries -- both assumed to span the tangent spaces at every point on a simply connected open set. Based on this equivalence, we build an explicit and convergent formula for the principal Koopman eigenfunctions defined on the region of attraction of a locally asymptotically stable equilibrium point, thereby offering a constructive formula to compute Koopman eigenfunctions.