Papers

The symmetric Macdonald polynomial at peculiar values of parameter $t=q^{-m}$, $m\in\mathbb{Z}_{\ge 0}$ is naturally split into non-symmetric parts, which are the (quasi)polynomial Baker-Akhiezer (BA) functions. One may think this is due to symmetricity, and one just picks up this way non-symmetric parts already containing all the information. However, we demonstrate that, in the case of non-symmetric Macdonald polynomials, it still works, though each single BA function splits into $N!$ distinct (quasi)polynomial BA functions. The sum of these functions gives rise to the universal solution of the eigenstate problem for the Cherednik Hamiltonians. Extending to arbitrary values of $t$ is also immediate giving rise to counterparts of the Noumi-Shiraishi power series. Altogether, this power series and its reductions to non-symmetric Macdonald polynomials and to BA functions form a non-symmetric triad. There are $N!$ different branches of the non-symmetric triad, each branch being split into $N!$ distinct triads, and of these $(N!)^2$ triads $N!(N-1)!$ are independent. We describe in detail the simplest $N=2$ case.

We consider integer-valued $\nabla\phi$ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions $d\ge3$, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.

We consider three specific $q$-deformed matrix models (the Chern-Simons, $q$-Laguerre, and $q$-Gaussian matrix models) which can be solved explicitly using the property of superintegrability. We show that one- and two-point functions of inverse characteristic polynomials can be interpreted as quantizations of Lagrangian subvarieties in $(\mathbb{C}^*)^2$ and $(\mathbb{C}^*)^4$, respectively. While such a geometric picture is expected for the Chern-Simons matrix model, our results for the $q$-Laguerre and $q$-Gaussian models are new and exhibit additional features. In particular, specific anti-symplectic birational involutions play an essential role in the construction. This reformulation provides a geometric framework for analyzing the semiclassical, large-$N$ expansion of matrix-model correlators. This picture is suggestive of the geometry underlying open topological strings, although the two constructions are not identical and the precise relation between them remains to be understood.

Motivated by mirror symmetry, we study exponential sums over the $\mathbb{F}_q$-points of the $A_1$ Coulomb branch. We show that their fiberwise structure is governed by polynomial Gauss sums, providing a geometric realization of Selberg character sums studied by Evans. Using Evans's evaluation, we obtain explicit formulas for the Coulomb-branch exponential sums as products of classical Gauss sums.

We prove that semi-strong convergence to a Bose–Einstein equilibrium below the critical temperature implies finite-time Bose–Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent $0\leq\eta<1$. For every admissible initial measure with $\overline T/\overline T_c<1$, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose–Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.

We introduce and study a new geometric representation for general classical spin models. It consists of two coupled percolation configurations that are the joint FK (random cluster) representation of the Ginibre rotation of two independent copies of the spin model, and can be viewed as an extension of Sheffield's cluster swapping defined in the context of height functions. Our approach combines the advantages of the random current and FK representations of the Ising model: it provides a percolation interpretation for various truncated correlation functions and at the same time satisfies the FKG inequality (for a large subclass of models). We highlight its strength and versatility by establishing two very different results. We first prove the converse of the classical fact that boundary multi-point correlation functions of planar Ising models are given by Pfaffians of the two-point functions. Indeed, we show that if a general spin model on a general graph satisfies the Pfaffian relations, then up to natural local modifications it must actually be an Ising model on a planar graph. In particular, the algebraic Pfaffian relations imply the topological feature of planarity. Our second application is a proof of sharpness of the phase transition for a class of spin models first considered by Ellis, Monroe and Newman, which we do by generalising the celebrated argument of Duminil-Copin and Tassion, replacing the use of the random current with our new representation.

We prove the Virasoro constraints for the relative Gromov–Witten theory of all smooth projective effective orbifold curves, allowing relative conditions at ordinary points. The absolute Jiang–Tseng Virasoro conjecture for smooth projective effective orbifold curves follows as a corollary.

We show that a superselction sector describing a single nonabelian anyon violates Haag duality, which is equivalent to the fundamental quantum-information theoretic principle of uniqueness of purifications. We provide a simple physics argument as well as a rigorous proof using sector theory, and instantiate our result concretely in Levin-Wen models. We also show that the associated ground state does not allow for quantum steering despite being a pure state. Finally, we show that this ground state violates approximate Haag duality, disproving the conjecture that all gapped ground states satisfy this condition. Our result implies that there are gapped phases of matter where Haag duality fails at every point in the phase.

Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system.

Non-parabolic growth of reaction layers is frequently observed in solid-state diffusion couples, but the relationship between microstructural evolution and the resulting kinetics remains incompletely understood. This effect is commonly attributed to grain growth, which can progressively reduce the contribution of fast grain-boundary diffusion, causing the effective diffusivity of a polycrystalline product layer to evolve during reaction. This work develops a moving-boundary diffusion model to describe product-layer growth while accounting for the local grain-growth history of the continuously formed product, as well as both bulk and grain-boundary diffusion. Numerical simulations show that the reaction can pass through three distinct kinetic regimes: an initial approximately parabolic regime dominated by grain-boundary diffusion, a transient sub-parabolic regime associated with strong spatial variation of the effective diffusivity, and a subsequent approximately parabolic regime dominated by bulk diffusion. The magnitude and duration of the sub-parabolic regime depend on the relative grain-boundary and bulk diffusivities and on the kinetics of grain growth. The instantaneous growth exponent therefore evolves continuously and does not represent a unique kinetic constant. Nevertheless, fitting simulated layer-thickness data over finite experimental time intervals produces well-defined apparent exponents, demonstrating how a transient process can appear to obey a single power law.

Pole-skipping occurs at special complex frequencies and momenta where the retarded Green function of a black-hole or black-brane background is not uniquely defined. The local near-horizon mechanism is well known: at resonance, a Frobenius recurrence matrix loses rank and the space of smooth horizon solutions enlarges. We address the global question of how two boundary-normalized solutions enter this resonant horizon solution space. For a general second-order scalar radial equation on a nonextremal background analytic near the horizon, the Frobenius recurrence at the resonant order yields a solvability condition. We prove that its vanishing is equivalent to singularity of the horizon recurrence matrix, absence of the logarithmic Frobenius term, and existence of two independent smooth horizon solutions. Away from resonance, the source zero is equivalent to horizon smoothness of the response-normalized solution, while the response zero is equivalent to horizon smoothness of the source-normalized solution. We then analyze the parameter-dependent simple pole of the nonresonant ingoing solution as resonance is approached. After removing this singularity, its resonant limit is proportional to the larger-root Frobenius solution, with proportionality factor given by the same solvability function. Continuing to the boundary shows that the same condition is equivalent to simultaneous vanishing of the two regularized boundary connection coefficients. This establishes a horizon-to-boundary dictionary between local Fuchsian resonance and the boundary source-response 0/0 structure of pole-skipping. Finally, different directions of approach can select different resonant solutions, with the selection governed by the first parameter variation of the same solvability function.

We formulate the type-I seesaw as a relativistic fermionic proximity effect in the exact active-sector two-point kernel. Active neutrinos have no renormalizable gauge-invariant Majorana mass and couple by lepton-number-conserving Yukawa hybridization to a sterile Majorana pairing kernel. Integrating out the sterile fields gives a nonlocal Nambu-Gorkov embedding self-energy. In a sterile Takagi basis its anomalous and normal flavor responses are $\mathcal{A}(Q^2)=\sum_i M_i y_i y_i^T/(Q^2+M_i^2)$ and $\mathcal{B}(Q^2)=\sum_i y_i y_i^\dagger/(Q^2+M_i^2)$, respectively. Their low-energy expansions generate correlated lepton-number-violating and lepton-number-conserving moment towers. The leading moments reproduce the Weinberg coefficient and the dimension-six kinetic operator. Both towers share one sterile mass set; its positive normal support supplies a common annihilator, although anomalous residues can cancel at exact degeneracy. We prove normal block-Hankel positivity and positivity of a dimensionally rescaled mixed Nambu-Hankel sequence. The latter gives matrix Cauchy-Schwarz inequalities between lepton-number-conserving and lepton-number-violating data and can reveal paired directions hidden by a degenerate normal residue. Isolated nondegenerate pole residues satisfy an additional nonlinear compatibility identity. These results yield tree-level EFT consistency tests and a matrix-pencil reconstruction of visible pole and aggregate-residue data; selecting a unique ultraviolet Lagrangian requires additional information about sterile-basis multiplicities. An exact one-generation example distinguishes the static Schur-complement kernel from the physical Takagi pole mass. The proximity structure applies without a Fermi surface or condensate, and Nambu doubling introduces no new physical degrees of freedom.

Non-collision singularities of the $n$-body problem are initial conditions for which no global solution exists and which, however, do not lead to a collision in the limit; i.e., the moment of inertia of the system diverges. The question of whether the set of initial conditions leading to non-collision singularities in the $n$-body problem is improbable is open and is the first on Barry Simon's list of fifteen problems in mathematical physics from the year 1984. So far, this question has only been answered positively in the case $n=4$. Using a final cluster decomposition into subsystems whose particles interact strongly with each other and exert only small forces on particles from different subsystems, we can prove the improbability of the set of all singular orbits with the following conditions: -There may be arbitrarily many subsystems undergoing total collision at their center of mass. - There may be subsystems exhibiting a non-collision singularity, i.e. the respective moment of inertia of the subsystem diverges. These subsystems consist of exactly four particles each and the asymptotic directions of these subsystems have to differ (which is only a mild condition). This result includes all known statements and indicates a way how the general problem could be solved.

We prove uniform-in-$N$ partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel–Riesz kernels on $\mathbb R^d$ in the locally square-integrable range $0\le s<d/2$. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal $N^{-1}$ normalized relative-entropy rate for the three-dimensional Coulomb flow; and an $N^{-1/2}$ Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman–Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.

We study finite-time singularities in $n$-body systems with pair potentials that are homogeneous of degree $-\alpha$, where $0<\alpha<2$, that are not necessarily attractive and with bounded time-dependent external forces. We develop a systematic method to analyse a colliding subsystem in the presence of simultaneous collision or non-collision singularities elsewhere in the system. We extend classical results of von Zeipel and Painlevé to such subsystems, establish bounds on their internal energy near total collision, and implement a McGehee blow-up that remains effective without conservation of energy. The blow-up yields the asymptotics of the cluster size, and in the case of attractive potentials, the convergence towards the set of central configurations and asymptotics of relative position, relative velocities, and internal angular momentum. As an application of the asymptotic results, we construct a simplified proof that the initial conditions leading to collision form a set of measure zero.

Conventional quantum-code constructions encode pure logical states as pure code states, but this restriction can sacrifice performance. We show that intrinsic encoding randomness can improve optimal entanglement fidelity. We bound the loss from imposing a rank-one encoder and prove it is at most quadratic near perfect recovery after joint optimization. The optimized advantage survives small noise perturbations. An explicit noise family requires higher-rank encoders arbitrarily close to perfect recovery, with every optimal encoder mapping pure inputs to mixed code states.

The quantum Rabi model occupies a distinguished role in the study of quantum light and matter, describing the most fundamental interactions. Elucidation of its spectral properties, along with those of its generalizations, is necessary for quantum optics and applications in areas such as quantum information technologies. In this paper, we explore a precise description of energy degeneracy of the two-photon asymmetric quantum Rabi model and the nature of the system's symmetries; as a result, we demonstrate that these have significant relationships with the structure of certain algebraic curves (hyperelliptic curves) and number theory. Concretely, we first prove the existence of degenerate eigenvalues using the monodromy data of the Fuchsian ODE, which draws the eigenvalue problem of the system, and the existence of associated symmetry operators, previously discovered and studied heuristically in physics. Subsequently, we give a complete characterization of spectral degeneracies and present a series of conjectures showing that the degeneracy may determine the entire energy spectral structure. More precisely, for sufficiently strong interaction, we obtain an excellent approximation of the energy curves by the zero locus of a two-variable polynomial associated to degenerate eigenstates. In addition, we postulate an explicit description of the relation between the missing symmetry at the Hamiltonian level (hidden symmetry) and the degeneracy of the two-photon quantum Rabi model.

We study a one-dimensional fractional $U(1)$ Ginzburg–Landau model whose quadratic part has Fourier multiplier $|k|^\sigma$, focusing on the marginal case $\sigma=1$. This dispersion yields logarithmic spin-wave fluctuations, suggesting BKT-like behavior despite the one-dimensional setting. We sample the equilibrium Gibbs measure using stochastic Gross–Pitaevskii dynamics and analyze correlation functions, dimensionless ratios, effective exponents, and twist responses. The correlation function shows a low-temperature algebraic branch with a temperature-dependent exponent, while the high-temperature regime exhibits a nonlocal-kernel-induced tail consistent with $C(r)\sim r^{-2}$. The correlation and Binder ratios are nearly size independent at low temperature and collapse with the BKT-type variable $(T-T_{\rm BKT})(\log L)^2$; finite-size effects set in around $T\simeq0.35\text{--}0.4$, consistent with $T_{\rm BKT}\simeq0.35$. Unlike the two-dimensional XY model, twist responses do not yield a finite helicity modulus: the ordinary linear-response quantity grows with system size, whereas the cusp twist response scales as $L^{-\eta(T)}$, like the squared zero-mode order parameter. Thus, the transition is BKT-like in correlation scaling, but lacks a universal helicity-modulus jump.

We introduce an operator-valued maximal f-divergence for completely positive (CP) maps between von Neumann algebras, associated with an operator convex function f on (0,+infinity). The construction takes values in the extended lower-semibounded self-adjoint part of the codomain von Neumann algebra. We prove independence of the common CP upper bound used in its computation, joint subadditivity, monotonicity under unital precomposition and normal postcomposition, a martingale convergence theorem for normal CP maps, and joint lower semicontinuity in the point-sigma-weak topology. For normal positive functionals, our construction recovers Hiai's maximal f-divergence. As a consequence, we establish its joint weak lower semicontinuity for arbitrary von Neumann algebras, answering a question left open by Hiai. For eta(t)=t log t, we obtain an operator-valued Belavkin–Staszewski (BS) relative entropy. Moreover, for normal channels with sigma-finite codomain, we prove that the BS channel divergence of Hollands and Ranallo coincides with the extended norm of our operator-valued divergence. Finally, we give finite-dimensional examples and obtain an explicit formula for a finite-index conditional expectation. In the BS case, this formula reduces to the logarithm of the Jones–Kosaki index.

We prove that the Hypergraph Neural Network, an invariant architecture with 3-body message passing, is a universal approximator for potential energy surfaces. Our main contribution is a multi-layer completeness theory. We show that $L$ layers of message passing on sparse, cutoff-based graphs achieve the same representational power as having access to the full $L$-hop neighborhood, provided the configurations are generic, satisfy an overlap condition and a connectivity condition. This provides the first rigorous justification for the common practice of using multi-layer message passing with a per-layer cutoff smaller than the physical interaction range, the setting used by virtually all practical graph neural network based machine-learned interatomic potentials. As immediate consequences, we show that both DPA3 and CHGNet architectures inherit universal approximation.

Recently in the physics literature there have been a small, but growing, number of papers that have applied the Eynard-Orantin topological recursion procedure to spectral curves with ramification loci of infinite cardinality. However, there is a corresponding gap in the mathematics literature; indeed, it has not been checked rigorously whether the resulting infinite sums converge, or whether the resulting correlators have all the desired properties of the topological recursion. The present work aims to bridge this gap by defining topological recursion on a suitably broad class of spectral curves to cover the aforementioned physics applications and to rigorously prove that this definition has the properties one would intuitively expect. As a by-product, some interesting coincidences involving quantum curves are discovered, where two different spectral curves yield the same quantum curve, a situation that has, hitherto, not appeared in the literature.

We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with $-\log\lambda_c$ locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Thus the critical-curve conjecture of Angel, de la Riva, Hermon, and Shi reduces to finiteness of the critical lifespan at every positive density. We identify the exact short-lifespan limit $t\lambda_c(t)\to\Delta\log(1/(1-p_c(G)))$, where $\Delta$ is the degree and $p_c(G)$ is the bond-percolation threshold, and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival. The comparison allows disconnected ranges and source-dependent laws, and therefore also applies to activation models beyond random walks on a fixed graph.

Motivated by deformations of spin structures and their geometric implications, we study a deformed exterior derivative, which is a third-order nilpotent operator. This higher-order differential structure induces an anholonomic rescaling of the local frame and leads to a bifurcation of the associated cohomological structure, with two distinct notions of closedness and exactness. We prove that the Poincaré Lemma fails in both sectors, so that closed deformed forms need not be exact even locally. We then apply the resulting framework to a four-dimensional effective field theory and show that a constant saturation of the deformation parameter can generate a hierarchy between the electroweak and fundamental Planck scales through a purely geometric mechanism, without introducing extra spatial dimensions.

Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\operatorname{Ind}(\mathcal{C}_G)$ of $\mathcal{C}_G$ is isomorphic to the algebra $\mathcal{O}(G/H)$ of regular functions on the homogeneous space $G/H$, for some reductive algebraic subgroup $H$ of $G$. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra $V_{\mathcal{L}_2}$ associated with the rank-one root lattice $\mathcal{L}_2:= \sqrt{2}\mathbb{Z}$, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra $L(1,0)$ with central charge $c=1$ satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of $L(1,0)$ satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational $c=1$ vertex operator algebras, up to the strong rationality of $V_{\mathcal{L}_2}^{\operatorname{A}_5}$.

We prove Conjecture 1.9 of Gabrielov-Novikov-Shapiro. More precisely, for every $p>0$, every nonconstant restriction to a line of a potential generated by $n$ point charges has at most $2n-1$ critical points. The bound is sharp, even for positive charges. The proof uses a duality with an auxiliary planar potential generated by collinear charges and Morse theory.

Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator's active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field $O(n)$-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions $U_i(S_i\cdot{S}_{i+1})$ and $V_i(S_i\cdot{S}_{i+2})$ is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler $O(n)$ open chain with nearest-neighbor interaction $V_i(\boldsymbol\sigma_i\cdot\boldsymbol\sigma_{i+1})$ and axial single-spin potential $U_i(\sigma_i^z)$ for every integer $n\ge1$ and every system size $L\ge1$. The homogeneous linear specialization maps the foundational frustrated $J_1$-$J_2$ model onto the canonical $J$-$h$ field model—with $n=1,2,3$ being the Ising, XY, and Heisenberg classical spin models, respectively; the theorem resolved a longstanding challenge for $n=3$ published in 1990. Its proof was done with an AI-synthesized recursive Householder moving frame and understood via a human-recognized hidden reciprocity. An analogous theorem holds when the continuous $O(n)$ spins are replaced by the $q$-state Potts spins, implying a closed-form exact solution of the $J_1$-$J_2$ standard Potts open chain for every $q\ge2$ and every $L\ge1$. The emergence of these theorems from a human-AI co-development framework suggests that sustained AI involvement throughout a systematic research program may incubate autonomous scientific breakthroughs and make aspects of the discovery process experimentally testable.

We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $\Theta=\alpha I_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbert matrix. For $\delta$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \[ \log \text{det}(I_N \pm H_N^\delta) = -\frac{\theta_\delta^2 \pm \theta_\delta}{2} \log N + O(1) \] where $\theta_\delta := \delta$ if $\delta < -1/2$, and $\theta_\delta := \frac 1 \pi \arcsin(\sin(\delta\pi))$ if $\delta \ge -1/2$ and $\arcsin \colon [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$ denotes the principal branch of $\arcsin$. For $\delta \geq -\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $\delta < - \frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $\delta < - \frac 12$. Most notably, we obtain for $\delta < -\frac12$ \[ \log \det\bigl(I_N - (H_N^\delta)^2\bigr) = -\delta^2 \log N + O(1). \] Such determinants arise in the study of Anderson's orthogonality catastrophe.

In this article, we establish a new dimensional reduction result in (Galilean or Lorentzian) Robertson Walker spacetimes with Euclidean fiber. In this context, we prove that every unchanged direction observer can be embedded into a three dimensional totally geodesic timelike submanifold.

We consider the problem of finding critical domains $\Omega\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (\Omega) = {\rm Per}\,(\Omega) + \frac 12 \iint_{\Omega\times \Omega } \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|\Omega|=V$. We look for smooth, embedded, compact surfaces $\partial\Omega$ that solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\frac 43}$.