The purpose of this work is to bring gravitational theories into play within the quickly developing framework of factorization algebras. We fit the causal structure of Lorentzian manifolds into categorical language, and in the globally hyperbolic case discover a convenient equivalence of coverages. Then, we show how both perturbative general relativity and perturbative conformal gravity define Batalin-Vilkovisky classical field theories. Finally, we describe how the observables of linearized general relativity define a particularly nice factorization algebra on the category of all globally hyperbolic manifolds and present a few conjectures which arise in specific cases, primarily motivated by the study of black hole entropy as a conserved Noether charge.

In rigorous study of stochastic models for the wave turbulence theory and R. Peierls's kinetic theory for the thermal conductivity in solids, analysis of integrals of the form $\int_{\mathcal{M}} \frac{F\omega_\mathcal{M}}{\Omega^2 + \nu^2\Gamma^2}$ and $\int_{\mathcal{M}} \frac{F\cos(\nu^{-1}\Omega)\omega_\mathcal{M}}{\Omega^2 + \nu^2\Gamma^2}$ plays a crucial role, where $\nu>0$ is a small parameter, $\mathcal{M}$ is a closed Riemannian manifold with volume form $\omega_\mathcal{M}$, and the functions $\Gamma > 0$, $F$, $\Omega$ are sufficiently smooth. We investigate the asymptotic behavior of the integrals in the limit $\nu\rightarrow 0$. This work continues studies [Kuksin' 17, Dymov' 23], in which the authors considered similar integrals for the case $\mathcal{M}=\mathbb{R}^d$ when the function $\Omega$ is Morse. We significantly weaken the latter assumption, which played an important role in the aforementioned works. This makes the obtained results applicable to the problem of rigorous justification of R. Peierls's kinetic theory.

Interest in non-reciprocally coupled systems recently led to the introduction of a minimal non-reciprocally coupled Cahn-Hilliard (CH) model by Brauns and Marchetti in 2024 arXiv:2306.08868, which we refer to as the Brauns-Marchetti (BM) model. This model can be seen as a conservative counterpart to the spatially extended FitzHugh-Nagumo model. Lacking a gradient structure, the BM model was observed to exhibit interesting dynamics including traveling periodic wave-trains and other coherent structures, as well as spatiotemporal chaos in certain parameter regimes. In this paper, we derive an effective equation for the interface dynamics of solutions to the BM model in $\mathbb{R}^2$ in the sharp-interface limit. The resulting system of equations is a generalization of the classical Mullins-Sekerka (MS) equations, which we refer to as the modified MS equations. We show that the modified MS equation shares some properties with its classical counterpart, but importantly, it is not in general a length minimizing flow. To illustrate the utility of this asymptotic reduction in the sharp interface limit, we perform a detailed analysis of stationary and periodic wave-trains, systematically deriving expressions for wave-train speeds and stability thresholds. The methods used here should be applicable to other non-reciprocally coupled CH models and therefore provide another avenue for their more detailed analysis.

We introduce the notion of a braided dynamical group which is a matched pair of dynamical groups satisfying extra conditions. It is shown to give a solution of the dynamical Yang-Baxter equation and at the same time a braided groupoid, thereby integrating the approaches of Andruskiewitsch and Matsumoto-Shimizu respectively that use these two notions to produce quiver-theoretical solutions of the Yang-Baxter equation. We pursue this connection further by relative Rota-Baxter operators on dynamical groups, which give rise to matched pairs of dynamical groups. As the derived structures of relative Rota-Baxter operators on dynamical groups, dynamical post-groups are introduced and are shown to be equivalent to braided dynamical groups. Finally, skew-braces are generalized to dynamical skew-braces as another equivalent notion of braided dynamical groups.

Relative entropy serves as a cornerstone concept in quantum information theory. In this work, we study relative entropy of random states from major generic state models of Hilbert-Schmidt and Bures-Hall ensembles. In particular, we derive exact yet explicit formulas of average relative entropy of two independent states of arbitrary dimensions from the same ensemble as well as from two different ensembles. One ingredient in obtaining the results is the observed factorization of ensemble averages after evaluating the required unitary integral. The derived exact formula in the case of Hilbert-Schmidt ensemble complements the work by Kudler-Flam (2021 Phys Rev Lett 126 171603), where the corresponding asymptotic formula for states of equal dimensions was obtained based on the replica method.

A method to construct a geometric structure with the same solutions as a given variational principle is presented. The method applies to large families of variational principles. In particular, the known results that assign cosymplectic geometry to Hamilton's principle and cocontact geometry to Herglotz's principle for regular Lagrangians are recovered. The unified Lagrangian-Hamiltonian formalism is also recovered via the absorption of the holonomy conditions. The method is applied to singular time-dependent Lagrangians, proving that they can always be described with a (pre)cosymplectic structure, although it is not always given by the Lagrangian $2$-form. When applied to singular action-dependent Lagrangians, the method does not always lead to (pre)cocontact geometry. In these cases, the resulting geometry associated with the Herglotz's variational principle is new.

We aim to characterise the spectral distributions of bi-infinite, semi-infinite, and finite aperiodic one-dimensional arrays of subwavelength resonators, constructed by sampling from a finite library of building blocks. By adopting the modern formalism of uniform hyperbolicity, we are able to strengthen and rigorously prove a Saxon-Hutner-type result, fully characterising the spectral gaps of the composite bi-infinite aperiodic system in terms of its constituent blocks. Crucial to this approach is a change of basis from transfer matrices to propagation matrices, allowing for a block-level characterisation. This approach also enables an explicit characterisation of edge-induced eigenmodes in the semi-infinite setting. Finally, we leverage finite section methods for Jacobi operators to extend our results to finite systems - providing strict bounds for their spectra in terms of their constituent blocks.

We propose a Molien--Weyl-type formula computing generating functions of invariants of supergroups $U(N|M)$, i.e. polynomials in supertraces, which arise as gauge groups of brane/negative brane systems in string theory. We either prove or numerically verify the formula in various examples. The formula further leads to a new expansion relating finite-$N$ and infinite-$N$ indices of $U(N)$ gauge theories. We comment on its relation to Murthy's Giant Graviton expansion, for which we suggest a physical interpretation in terms of ``Koszul dual" branes and negative branes.

We study the laws of the two-dimensional vector-valued Dirichlet Gaussian free field and its massive lattice counterpart, conditioned to avoid a ball at every site of a subdomain. We prove that, under this conditioning, the norm of the massless field exhibits entropic repulsion, while its angular components freeze at all mesoscopic scales. A key step in the analysis is showing that around any given point in the bulk of the range, the unconditioned field has no holes. In the massive case, the conditioned field behaves differently: its norm remains uniformly bounded as the system size grows, leading to the existence of infinite-volume Gibbs measures. Furthermore, in the scalar massive case, the system undergoes a phase transition in the size of the avoided interval: for small intervals, the system admits a unique infinite-volume limit, while for sufficiently large centered intervals, multiple such limits exist.

Quantum Monte Carlo method with operator-loop update is a powerful technique that has been extensively used with great success in condensed matter physics. It enables one to sample from thermal and ground states of local Hamiltonians of various spin, bosonic and fermionic systems as long as the Hamiltonian does not have a negative-sign problem. Despite the practical success of this method, theoretical understanding of the efficiency of the algorithm has been lacking. The operator-loop update is commonly used for path-integral formulation (Suzuki-Trotter/world-lines) of the partition function. In this work we consider this method applied to the stoquastic (sign-problem free) XY model and prove that the mixing time of the Markov chain is polynomial in the system size and the inverse temperature. Using the fast mixing Markov chain, we can estimate the partition functions of the Hamiltonians that we consider in a polynomial time, significantly improving upon the best known previous algorithm by Bravyi and Gosset [arXiv:1612.05602]. Our algorithm also allows for natural extensions to a wide class of empirically fast-mixing Hamiltonians.

In this note we use an approximation scheme to establish large deviations for quasi-periodic Gevrey cocycles. As an application, we obtain continuity in the cocycle for the Lyapunov exponent.

There is considerable current interest in applications of generalised Lie algebras graded by an abelian group $\Gamma$ with a commutative factor $\omega$. This calls for a systematic development of the theory of such algebraic structures. We treat the representation theory and invariant theory of the $\Gamma$-graded general linear Lie $\omega$-algebra $\mathfrak{gl}(V(\Gamma, \omega))$, where $V(\Gamma, \omega)$ is any finite dimensional $\Gamma$-graded vector space. Generalised Howe dualities over symmetric $(\Gamma, \omega)$-algebras are established, from which we derive the first and second fundamental theorems of invariant theory, and a generalised Schur-Weyl duality. The unitarisable $\mathfrak{gl}(V(\Gamma, \omega))$-modules for two ``compact'' $\ast$-structures are classified, and it is shown that the tensor powers of $V(\Gamma, \omega)$ and their duals are unitarisable for the two compact $\ast$-structures respectively. A Hopf $(\Gamma, \omega)$-algebra is constructed, which gives rise to a group functor corresponding to the general linear group in the $\Gamma$-graded setting. Using this Hopf $(\Gamma, \omega)$-algebra, we realise simple tensor modules and their dual modules by mimicking the classic Borel-Weil theorem. We also analyse in some detail the case with $\Gamma={\mathbb Z}^{\dim{V(\Gamma, \omega)}}$ and $\omega$ depending on a complex parameter $q\ne 0$, where $\mathfrak{gl}(V(\Gamma, \omega))$ shares common features with the quantum general linear (super)group, but is better behaved especially when $q$ is a root of unity.

We study ergodic theoretical properties of flows on circle bundles over translation surfaces that arise via prequantization, generalizing the theory of Heisenberg nilflows to base surfaces more general than tori; these flows are among the most fundamental examples of parabolic dynamical systems with non-trivial central directions. In particular, we show that such flows are relatively mixing, i.e., they exhibit decay of correlations in the orthogonal complement of functions constant along fibers. We discuss applications of this result to the dynamics of such flows, to the ergodic theory on the corresponding space of wave functions, and, via surface of section constructions, to the study of affine skew products over interval exchange transformations, in the spirit of Furstenberg's classification program for measurable dynamical systems.

Quantum entanglement is a useful resource for implementing communication tasks. However, for the resource to be useful in practice, it needs to be accessible by parties with bounded computational resources. Computational entanglement measures quantify the usefulness of entanglement in the presence of limited computational resources. In this paper, we analyze systematically some basic properties of two recently introduced computational entanglement measures, the computational distillable entanglement and entanglement cost. To do so, we introduce lower bound and upper bound extensions of basic properties to address the case when entanglement measures are not defined by a scalar value but when only lower or upper function bounds are available. In particular, we investigate the lower bound convexity and upper bound concavity properties of such measures, and the upper and lower bound additivity with respect to the tensor product. We also observe that these measures are not invariant with local unitaries, although invariance is recovered for efficient unitaries. As a consequence, we obtain that these measures are only LOCC monotones under efficient families of LOCC channels. Our analysis covers both the one-shot scenario and the uniform setting, with properties established for the former naturally extending to the latter.

We consider multiple radial SLE curves with various time parameterizations and possible spiraling behavior. We construct them by tilting independent radial SLEs with a suitable local martingale, generalizing the earlier construction by Healey and Lawler. We prove that the curves are almost surely transient (i.e., they emanate from boundary points and terminate at a common interior target point). We show that they enjoy the resampling property: conditional on all of the curves but one, the remaining curve is distributed as chordal SLE in the remaining domain. We also verify that the multiradial SLE measure satisfies a natural boundary perturbation property analogous to that of the known SLE variants, involving its partition function (which is finite). Interestingly, in the parlance of Coulomb gas formalism in conformal field theory, partition functions of multiradial SLE processes with spiral involve both electric and magnetic charges.

Gauging and duality transformations, two of the most useful tools in many-body physics, are shown to be equivalent up to constant depth quantum circuits in the case of one-dimensional quantum lattice models. This is demonstrated by making use of matrix product operators, which provide the lattice representation theory for global (categorical) symmetries as well as a classification of duality transformations. Our construction makes the symmetries of the gauged theory manifest and clarifies how to deal with static background fields when gauging generalised symmetries.

We investigate the asymptotic equipartition property (AEP) in the context of multipartite entanglement measures on pure states. Specifically, we formulate AEP for subadditive entanglement measures that admit certain weak conditions. This is motivated by the uniqueness of the entanglement entropy in the asymptotic limit in the bipartite case. On the other hand, its operational relevance comes from the $\text{LOCC}_q$ scenario (asymptotic local operations and classical communication with a sublinear amount of quantum communication). Analogously to the classical AEP, we prove that the regularization of smooth weakly additive entanglement measures (subject to some weak extra conditions) yields weakly additive and asymptotically continuous entanglement measures. Then evaluate the mentioned regularization and smoothing on known Rényi type multipartite entanglement measures, showing that the resulting regularized entanglement measures reduce to convex combinations of bipartite entanglement entropies.

We study translation-invariant quantum spin Hamiltonians on general graphs with non-commuting interactions either given by (i) a random rank-$1$ projection or (ii) Haar projectors. For (i), we prove that the Hamiltonian is gapped on any bounded-degree graph with high probability at large local dimension. For (ii), we obtain a gap for sufficiently large local dimension. Our results provide examples where the folklore belief that typical translation-invariant Hamiltonians are gapped can be proved, which extends a result by Bravyi and Gosset from 1D qubit chains with rank-$1$ interactions to general bounded-degree graphs. We derive the gaps by analytically verifying generalized Knabe-type finite-size criteria that apply to any bounded-degree graph.

The Lagrangian-Averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$) model, a turbulence closure scheme based on energy-conserving modifications to nonlinear advection, can produce more energetic simulations than standard models, leading to improved fidelity (e.g., in ocean models). However, comprehensive understanding of the mechanism driving this energetic enhancement has proven elusive. To address this gap, we derive the fast quasi-geostrophic limit of the three-dimensional, stably-stratified LANS-$\alpha$ equations. This provides both the slow, balanced flow and the leading-order fast wave dynamics. Analysis of these wave dynamics suggests that an explanation for the energetic enhancement lies in the dual role of the smoothing parameter itself: increasing $\alpha$ regularizes the dynamics and simultaneously generates a robust landscape of wave-wave resonant interactions. Direct numerical simulations show that $\alpha$ plays an analogous role to that of the Burger number ($Bu$) in governing the partition of energy between slow and fast modes -- and consequently, the timescale of geostrophic adjustment -- but with key differences. Increasing $\alpha$, regardless of the relative strengths of rotation and stratification, extends the lifetime of wave energy by delaying the dominance of the slow modes. We find that the creation of an energy pathway only involving fast waves is a universal outcome of the regularization across all values of $Bu$, contrasting with a disruption of slow-fast interactions that is most impactful only in the $Bu=1$ case. These insights unify the LANS-$\alpha$ model's characteristic energetic enhancement with, in some cases, its known numerical stiffness, identifying potential pathways to mitigate stability issues hindering the broader application of LANS-$\alpha$-type models.

We provide a systematic construction for local quantum circuits hosting free fermions in disguise, both with staircase and brickwork architectures. Similar to the original Hamiltonian model introduced by Fendley, these circuits are defined by the fact that the Floquet operator corresponding to a single time step can not be diagonalized by means of any Jordan-Wigner transformation, but still displays a free-fermionic spectrum. Our construction makes use of suitable non-local transfer matrices commuting with the Floquet operator, allowing us to establish the free fermionic spectrum. We also study the dynamics of these circuits after they are initialized in arbitrary product states, proving that the evolution of certain local observables can be simulated efficiently on classical computers. Our work proves recent conjectures in the literature and raises new questions on the classical simulability of free fermions in disguise.

Given a quantum state in the finite-dimensional Hilbert space $ \C^n $, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with: (i) an ``ortho-measurable'' function defined on the Boolean ``ortho-algebra'' generated by the eigenspaces that form an orthogonal decomposition of $ \C^n $; (ii) a ``numerically identified'' orthogonal decomposition of $ \C^n $. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on $ \C^n $ is identified with a Bayesian prior ``ortho-probability'' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule.

The quantum Schur transform has become a foundational quantum algorithm, yet even after two decades since the seminal 2005 paper by Bacon, Chuang, and Harrow (BCH), some aspects of the transform remain insufficiently understood. Moreover, an alternative approach proposed by Krovi in 2018 was recently found to contain a crucial error. In this paper, we present a corrected version of Krovi's algorithm along with a detailed treatment of the high-dimensional version of the BCH Schur transform. This high-dimensional focus makes the two versions of the transform practical for regimes where the number of qudits $n$ is smaller than the local dimension $d$, with Krovi's algorithm scaling as $\widetilde{O}(n^4)$ and BCH as $\widetilde{O}(\min(n^5,nd^4))$. Our work addresses a key gap in the literature, strengthening the algorithmic foundations of a wide range of results that rely on Schur--Weyl duality in quantum information theory and quantum computation.

Inverse magnetisation problem consists in inferring information about a magnetic source from measurements of its magnetic field. Unlike a general magnetisation distribution, the total magnetisation (net moment) of the source is a quantity that theoretically can be uniquely determined from the field. At the same time, it is often the most useful quantity for practical applications (on large and small scales) such as detection of a magnetic anomaly in magnetic prospection problem or finding the overall strength and mean direction of the magnetisation distribution of a magnetised rock sample. It is known that the net moment components can be explicitly estimated using the so-called Helbig's integrals which involve integration of the magnetic field data on the plane against simple polynomials. Evaluation of these integrals requires knowledge of the magnetic field data on a large region or the use of ad hoc methods to compensate for the lack thereof. In this paper, we derive higher-order analogs of Helbig's integrals which permit estimation of total magnetisation components in terms of measurement data available on a smaller region. Motivated by a concrete experimental setup for analysing remanent magnetisation of rock samples with a scanning microscope, we also extend Helbig's integrals to the situation when knowledge of only one field component is necessary. Moreover, apart from derivation of these novel formulas, we rigorously prove their accuracy. The presented approach, based on an appropriate splitting in the Fourier domain and estimates of oscillatory integrals (involving both small and large parameters), elucidates the derivation of asymptotic formulas for the net moment components to an arbitrary order, a possibility that was previously unclear. The obtained results are illustrated numerically and their robustness with respect to the noise is discussed.

The surface charges associated to $p$-form gauge fields in arbitrary spacetime dimension for large values of the radial coordinate are computed. We show that, under the Hodge duality between the field strength of the dual formulations, electric-like charges for $p$-forms are mapped to magnetic-like charges for the dual $q$-forms, with $q = D-p-2$. We prove an existence and uniqueness theorem for the duality map linking asymptotic electric-like charges of the dual descriptions and we give it an algebraic topology interpretation. As a result the duality map has a topological nature and ensures the charge of a description has information of the dual description leading to a deeper understanding of gauge theories, of the non-trivial charges associated to them and of the duality of their observables. Moreover a link between higher form symmetry charges, naturally associated to a $p$-form gauge theory, and their asymptotic charges is proposed. The higher form charges are reproduced choosing the gauge parameter to be constant and to have support only on an appropriate codimension submanifold. This could partially answer to an open question of the celestial holography program.

We provide a new solution to the classical black box problem (the discrete Calderon problem) in the theory of circular electrical networks. Our approach is based on the explicit embedding of electrical networks into non-negative Grassmannians and generalized chamber ansatz for it. Also, we reveal the relation of this problem with the combinatorial properties of spanning groves and the theory of totally non-negative matrices. Key words: electrical networks, discrete Calderon problem, discrete electrical impedance tomography, non-negative Grassmannians, twist for positroid variety, Temperley trick, totally non-negative matrices, effective resistances.

We study heat conduction in a one-dimensional finite, unpinned chain of atoms perturbed by stochastic momentum exchange and coupled to Langevin heat baths at possibly distinct temperatures placed at the endpoints of the chain. While infinite systems without boundaries are known to exhibit superdiffusive energy transport described by a fractional heat equation with the generator $-|\Delta|^{3/4}$, the corresponding boundary conditions induced by heat baths remain less understood. We establish the hydrodynamic limit for a finite chain with $n+1$ atoms connected to thermostats at the endpoints, deriving the macroscopic evolution of the averaged energy profile. The limiting equation is governed by a non-local Lévy-type operator, with boundary terms determined by explicit interaction kernels that encode absorption, reflection, and transmission of long-wavelength phonons at the baths. Our results provide the first rigorous identification of boundary conditions for fractional superdiffusion arising directly from microscopic dynamics, highlighting their distinction from both diffusive and pinned-chain settings.

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessary real. In most cases, the complexification of those Lie algebra is necessary in order to derive irreductible representations of the Lie algebra and subsequently of the symmetry group. In this paper, we give a precise definition of the concept and prove step by step an important result $$\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}. $$ This result is used to determine the irreductible representations of the proper Lorentz group and thus the physical objects admissible when this symmetry is present. It is shown that finite representations of the proper Lorentz group are characterized by pairs of half-integers $(j_1,j_2)$, which determine unambiguously the physical object associated to the given representation. For example, the representation $(0,0)$ of dimension $1$ is called the scalar representation, it corresponds to the Higgs field, and $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ of dimension $4$ is called the Dirac spinor representation, it corresponds to matter particle called fermions. This means that the mathematical group structure determines the material content of the universe following this algebraic structure.

Quantum teleportation can be used to define a notion of parallel transport which characterizes the entanglement structure of a quantum state \cite{Czech:2018kvg}. This suggests one can formulate a gauge theory of entanglement. In \cite{Wong:2022mnv}, it was explained that measurement based quantum computation in one dimension can be understood in term of such a gauge theory (MBQC). In this work, we give an alternative formulation of this "entanglement gauge theory" as an extended topological field theory. This formulation gives a alternative perspective on the relation between the circuit model and MBQC. In addition, it provides an interpretation of MBQC in terms of the extended Hilbert space construction in gauge theories, in which the entanglement edge modes play the role of the logical qubit.

We calculate the transition rate of an Unruh-DeWitt detector coupled to a non-equilibrium steady state (NESS) of a free massless scalar field on four-dimensional Minkowski spacetime. Bringing two semi-infinite heat baths at different temperatures into thermal contact along a surface, the NESS arises at asymptotically late times as a stationary state that has modewise thermal properties and features a heat flow between the reservoirs. The detector couples linearly to the field by a monopole interaction, and it moves inertially along the axis of the NESS heat flow. We contrast the transition rate with the case of a detector that is coupled to an inertial thermal equilibrium state. The results illustrate that the monopole does not couple to the heat flow, causing the detector to only register kinematical effects. Hence dynamical features of the NESS are hidden from this detector model.

The one-dimensional (1D) $J_1$-$J_2$ $q$-state Potts model is solved exactly for arbitrary $q$ by analytically block-diagonalizing the original $q^2\times q^2$ transfer matrix into a simple $2\times 2$ maximally symmetric subspace, based on using OpenAI's reasoning model o3-mini-high to exactly solve the $q=3$ case. Furthermore, by matching relevant subspaces, we map the Potts model onto a simpler effective 1D $q$-state Potts model, where $J_2$ acts as the nearest-neighbor interaction and $J_1$ as an effective magnetic field, nontrivially generalizing a 56-year-old theorem previously limited to the simplest case ($q=2$, the Ising model). Our exact results provide insights to phenomena such as atomic or electronic order stacking in layered materials and the emergence of dome-shaped phases in complex phase diagrams. This work is anticipated to fuel both research in 1D frustrated magnets for recently discovered finite-temperature application potentials and the fast moving topic area of AI in science.

We studied global density-of-states correlation function $R(\omega)$ for Lévy-Rosenzweig-Porter random matrix ensemble in the non-ergodic extended phase. Using an extension of Efetov's supersymmetry approach we calculated $R(\omega)$ exactly in all relevant ranges of $\omega$. At relatively low $\omega \leq \Gamma$\, (with $\Gamma \gg \Delta$ being the effective miniband width) we found GUE-type oscillations with period of level spacing $\Delta$, decaying exponentially at the Thouless energy scale $E_{Th} = \sqrt{\Delta \Gamma/2\pi}$. At high energies $\omega \gg E_{Th}$ our results coincide with those obtainen via cavity equation approach. Inverse of the effective miniband width, $1/\Gamma$, is shown to be given by the average of the local decay times over Lévy distribution.

Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to 2-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: 2-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker -- yet still non-trivial -- universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.

In this paper we explore ideas of holography and strings living in the $d+1$ dimensional Anti-de Sitter space $AdS_{d+1}$ in a unified framework borrowed from twistor theory. In our treatise of correspondences between geometric structures of the bulk $AdS_{d+1}$, its boundary and the moduli space of boundary causal diamonds aka the kinematic space ${\mathbb K}$, we adopt a perspective offered by projective geometry. From this viewpoint certain lines in the $d+1$ dimensional real projective space, defined by two light-like vectors in ${\mathbb R}^{d,2}$ play an important role. In these projective geometric elaborations objects like Ryu-Takayanagi surfaces, spacelike geodesics with horospheres providing regularizators for them and the metric on ${\mathbb K}$ all find a natural place. Then we establish a correspondence between classical strings in $AdS_{d+1}$ and causal diamonds of its asymptotic boundary. At each point on the worldsheet, the tangent vectors $\partial_\pm X$ are projected onto boundary coordinates that identify the past and future tips of a causal diamond. Under this projection, the string equations of motion translate into a dynamics of boundary causal diamonds. A procedure for lifting up a causal diamond to get a proper string world sheet is also developed. In this context we identify an emerging $SO(1,1)\times SO(1,d-1)$ gauge structure incorporated into a Grassmannian $\sigma$-model targeted in ${\mathbb K}$. The $d=2$ case is worked out in detail. Surprisingly in this case $AdS_3$ with its strings seems to be a natural object which is living inside projective twistor space. On the other hand ${\mathbb K}$ (comprising two copies of two dimensional de Sitter spaces) is a one which is living inside the Klein quadric, as a real section of a complexified space time.

Systems undergoing phase-ordering kinetics after a quench into the ordered phase with $0<T<T_c$ from a fully disordered initial state and with a non-conserved order-parameter have the dynamical exponent ${z}=2$. The long-time behaviour of their single-time and two-time correlators, determined by the noisy initial conditions, is derived from Schrödinger-invariance and we show that the generic ageing scaling forms of the correlators follow from the Schrödinger covariance of the four-point response functions. The autocorrelation exponent $\lambda$ is related to the passage exponent $\zeta_p$ which describes the time-scale for the cross-over into the ageing regime. Both Porod's law and the bounds $d/2 \leq \lambda \leq d$ are reproduced in a simple way. The dynamical scaling in fully finite systems and of global correlators is found and the low-temperature generalisation $\lambda= d-2\Theta$ of the Janssen-Schaub-Schmittmann scaling relation is derived.

The Gutzwiller trace formula relates the asymptotic spacing of quantum-mechanical energy levels in the semiclassical limit to the dynamics of periodic classical particle trajectories. We generalize this result to the case of non-smooth potentials, for which there is partial reflection of energy from derivative discontinuities of the potential. It is the periodic trajectories of an associated branching dynamics that contribute to the trace asymptotics in this more general setting; we obtain a precise description of their contribution.

In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius $b$. Instead of the usual case of the refractive index $\rho\in W^2_2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $\rho$ is a piecewise $ W^1_2$ function. We prove that if $\int_0^b \sqrt{\rho(r)} dr<b$, then $\rho$ is uniquely determined by all special transmission eigenvalues; if $\int_0^b \sqrt{\rho(r)} dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $\rho$. We also consider the mixed spectral problem and obtain that $\rho$ is uniquely determined from partial information of $\rho$ and the ``almost real subspectrum".