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Wednesday, 4 November 2026: 11 new papers, 26 cross-lists and 39 replacements in Mathematical Physics (math.MP), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real $n$-dimensional vector space $V$, which is denoted by $\mathcal{C}l\left(H_{V}\right)$. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra $\mathcal{C}l(H_V)$ simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework---known as the ``mother algebra''---for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space $H_{V}$ are presented. The Clifford--Lipschitz group and the Pin and Spin groups associated with $\mathcal{C}l\left(H_{V}\right)$ are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived.
Nonlinear lattice kinetic simulation requires each updated distribution to supply the velocity and gradients that determine the next step. In a quantum-classical solver, reading out the full distribution incurs work proportional to the lattice size. We construct a tensor-train (TT) interface that retains reusable classical cores at the physical amplitude scale for a full-relaxation lattice kinetic scheme. The cores supply nonlinear coefficients and prepare each quantum input. Energy-scaled propagation and coordinate recovery encode the updated distribution, and alternating least-squares reconstruction converts signed Hadamard-test overlaps into output cores by restoring a known normalization factor. The next step starts from accepted cores without replaying previous quantum evolution. Under positive, unit-sum reference coefficients, positive lower and finite upper density bounds, and Lipschitz equilibrium sensitivity, perturbation bounds relate local defects to feedback errors. Resource bounds account for classical operations, circuit execution, sampling, and verification. Classical diagnostics expose why distribution accuracy alone is insufficient: in 80-step feedback on a $16\times16$ grid, rank cap 4 nearly eliminates the vortex despite a final distribution error below $1\%$, whereas rank cap 12 gives relative velocity error $1.03\times10^{-3}$. Four-step binomial readout simulations show that tightening core sampling tolerance threefold raises the prescribed sufficient budget from $5.97\times10^{12}$ to $5.37\times10^{13}$ shots as the result approaches the fixed-rank, fixed-sweep exact-overlap trajectory. The interface therefore provides a way to connect quantum propagation to nonlinear feedback and to assess the separate demands of compression, velocity accuracy, and readout precision.
Following some recent results, we consider here a particular class of states on a $O^\ast$-algebra $\M$ which we call {\em eigenstates of an element} $A\in\M$, and we deduce some of their properties. Then, we apply our definition to a family of ladder elements, i.e. elements of $\M$ obeying certain algebraic (commutation) relations which are physically motivated, and we discuss several results for these states
We revisit the classical spectral problem for the Laplace-Beltrami and Laplace-Bochner operators on complex Grassmannians. After reviewing the standard approach based on non-commutative harmonic analysis, we develop two complementary methods originating from the quantization of the corresponding one-dimensional sigma model, both with and without a magnetic monopole field. The first is based on $\mathbf{SU}(n)$-equivariant first-order GLSM formulation and gives explicit eigenfunctions in terms of projectors and Pl\"ucker coordinates. The second uses a spin-chain realization of free particles on manifolds and describes the space of square-integrable functions, or sections in the magnetic case, through limiting tensor products of long rectangular $\mathbf{SU}(n)$ representations. Together, these approaches provide a more explicit picture of the spectral geometry of Grassmannians and clarify the relation between harmonics, Young diagrams, magnetic bundles, and Laplacian spectra.
We construct a measure on continuous paths in the future cone of two-dimensional space that is invariant under the orthochronous Lorentz group and quasi-invariant under the action of the diffeomorphism group. We use the proposed measure to obtain the fundamental solution of the heat equation. The result agrees with the expression obtained by other methods.
Self-similar solutions play an important role in the formation of singularities in the Yang--Mills equations. In five space dimensions, these equations admit infinitely many self-similar solutions with an increasing number of unstable modes. It was observed numerically that all excited profiles share the least unstable eigenvalue $\lambda=5$. We explain this remarkable spectral property by an elementary construction of the corresponding eigenfunction in terms of the profile.
Hirota's three-soliton condition is the standard test for the integrability of a bilinear equation $P(D)f\cdot f=0$, and the folklore implication ``three-soliton condition $\Rightarrow$ $N$-soliton solutions for all $N$'' underlies the classifications of Hietarinta. We study this implication in two directions. The first is the family $L\,(D_x^3+\Lambda)^{2n+1}f\cdot f=0$ with $L$, $\Lambda$ linear forms in the Hirota operators, which for $n=0$ contains the Korteweg--de Vries, Ito and Boiti--Leon--Manna--Pempinelli equations and for $L=D_x$ is the family of G\"urses and Pekcan. On the principal component of the dispersion variety the $N$-soliton condition reduces to a sign sum of two bialternants; counting odd exponents shows that for $n=0$ the class is Hirota integrable and that for $n\ge1$ the three-soliton condition holds for every $L$, and an explicit formula for the four-soliton residual proves the G\"urses--Pekcan conjecture that the four-soliton condition fails for every $n\ge1$. The formula yields sum-of-squares certificates excluding four-soliton solutions at every real configuration for the G\"urses--Pekcan and Ito equations, classifies the real members of the class, and determines maximal numbers of nondegenerate solitons: exactly four or five for $n=1$ according as $L$ lies in the span of $D_x,\Lambda$ or not, five for $n=2$ and $L=D_x$, and finite for every $n\ge1$ and every $L$. The second direction is a reformulation of the implication for a single non-hyperplane component of multiplicity one, which we prove, with complete classifications, for all graph-type dispersions $L(q(D)+D_t)$, for every irreducible even quadric component in any dimension and with an arbitrary cofactor, for parabolic components, for the dispersions $F(\ell)+Q$ containing the Kadomtsev--Petviashvili surface, and, with a prescribed one-parameter phase family, for the cubic graph.
In this paper, a compact high-order gas-kinetic scheme (HGKS) is developed for compressible arbitrary Lagrangian-Eulerian (ALE) computations on three-dimensional unstructured meshes. To achieve high-order spatial accuracy, a large number of matrix inversions are required for spatial reconstruction. In ALE computations, the mesh moves and the geometric information must be recalculated at each time step, which reduces computational efficiency. To improve efficiency, a memory-reduction reconstruction strategy is employed to construct a quadratic polynomial on unstructured meshes. By differentiating the candidate polynomial, the quadratic terms are obtained by the least-squares method using the average gradient values of the cell itself and its neighboring cells. The linear terms of the polynomial are also determined by the least-squares method. On three-dimensional unstructured meshes, a $3\times3$ coefficient matrix must be stored, and four least-squares procedures are required to construct a quadratic polynomial. To suppress numerical oscillations in flows with discontinuities, generalized essentially non-oscillatory (GENO) reconstruction and a gradient compression factor are adopted. To accelerate the computation, a multi-GPU accelerated code is implemented using Compute Unified Device Architecture (CUDA) and Message Passing Interface (MPI). The current scheme is fully explicit, and the acceleration framework can be developed based on cell-level parallelism. Numerical examples are presented to evaluate the accuracy and robustness of the scheme, including flows with strong discontinuities, pitching wings, and helicopter rotors. Numerical results confirm the excellent performance of the compact ALE scheme on unstructured meshes.
In this paper, we propose a method based on rasterization in computer graphics for the integration of discontinuous functions accurately and efficiently on the GPU. The proposed method takes advantage of the massively parallel architecture of GPUs via a modified graphics pipeline, where discontinuities are encoded through geometric primitives and processed in the rasterization stage, and multisample anti-aliasing (MSAA) is used to improve integration accuracy across subdomain interfaces. We present both hardware- and software-based pipelines and compare their performance in terms of accuracy as well as computational effort on modern GPU hardware with adaptive quadrature on the CPU. For the tested hardware, at approximately comparable median relative error, the hardware pipeline is up to around $100\text{x}$ faster than adaptive quadrature, while the single- and double-precision software pipelines achieve up to $840\text{x}$ and $1200\text{x}$ speedups compared to their adaptive quadrature counterparts. We use the finite cell method as model problem, where cutcells represent discontinuous integrands at the interface of the physical and fictitious domains and demonstrate the performance of the proposed method using numerical benchmarks. The end-to-end finite cell solution runtime is reduced by a factor of $8.3$ to $9.5$ in the single-precision pipeline and by a factor of $62$ to $72$ in the double-precision pipeline at the prescribed integration settings. Therefore, results indicate that the rasterization-based integration method is a promising alternative for the accurate and efficient integration of discontinuous functions in demanding applications on the GPU.
We consider the one-dimensional Dirac operator $D_{A,c}$ with a general, not necessarily self-adjoint, point interaction at the origin, whose strength is described by an arbitrary complex $2\times 2$ matrix $A$ and suitably rescaled by the speed of light $c$, together with its non-local approximations $D_{A,c}^{\epsilon}$ obtained by replacing the Dirac distribution by a rank-one projection onto a scaled function $v_\epsilon$. It is known that $D_{A,c}^\epsilon \to D_{A,c}$ as $\epsilon\to 0+$ and that $D_{A,c}-c^2/2$ converges, as $c\to+\infty$, to the Schr\"odinger operator $H_A$ with the general point interaction, both in the norm resolvent sense. We prove that the first convergence is uniform with respect to $c$ in a neighbourhood of $+\infty$, and we provide explicit rates of convergence. Combining this with an explicit computation of the non-relativistic limit of $D_{A,c}^\epsilon$ for fixed $\epsilon$, we show that the two limits can be interchanged and that the corresponding double limit exists. As a by-product we obtain a family of non-local, rank-two approximations $H_A^\epsilon$ of $H_A$ that converge to $H_A$ in the norm resolvent sense. Contrary to the relativistic setting, the coupling constants of $H_A^\epsilon$ are renormalized; we give an explicit formula for this renormalization.
This article studies the dynamical properties of the multiple elastic pendulum formed by an arbitrary number of point masses connected in a chain by elastic springs. First, using the Lagrangian formulation, the equations of motion governing the system's behavior, accounting for both large-angle displacements and elastic deformations, are derived. It is shown that well-established, low-dimensional systems, such as the classical double elastic pendulum, can be systematically recovered as particular cases directly from the general equations of motion after applying the appropriates particular conditions. The behavior of the system under small oscillations near its stable equilibrium is also investigated. For this case, the linearized equations allow the solution of the eigenvalue problem, demonstrating that in this case uncoupled behavior between the angular displacements and the spring elongations.
Uncertainty quantification in PDE inverse problems is essential in many applications. Scientific machine learning and AI enable data-driven learning of model components while preserving physical structure, and provide the scalability and adaptability needed for emerging imaging technologies and clinical insights. We develop a Bilevel Local Operator Learning framework for Bayesian inference in PDEs (B-BiLO). At the upper level, we sample parameters from the posterior via Hamiltonian Monte Carlo, while at the lower level we fine-tune a neural network via low-rank adaptation (LoRA) to approximate the solution operator locally. B-BiLO enables efficient gradient-based sampling without synthetic data or adjoint equations and avoids sampling in high-dimensional weight space, as in Bayesian neural networks, by optimizing weights deterministically. We analyze errors from approximate lower-level optimization and establish their impact on posterior accuracy. Numerical experiments across PDE models, including tumor growth, demonstrate that B-BiLO achieves accurate and efficient uncertainty quantification.
The Thermodynamic Bethe Ansatz (TBA) procedure developed by Yang-Yang provides the free energy functional to be minimized via a variational problem. Their result not only gives the correct thermodynamics, but it also sets the framework for Generalized Hydrodynamics. Originally developed for quantum systems, this procedure has been recently successfully applied to (integrable) classical particle systems. However, this has never been done for classical field theories: no classical derivation has consistently accounted for solitons, breathers, radiation (dispersive modes), and their mutual interactions. We here derive the TBA equations for the Toda chain directly from its inverse scattering transform. This results in solitonic modes interacting with radiation, extending the soliton gas picture. We provide numerical verification of our equations against well established TBA equations in terms of particle modes. We expect the same procedure should apply to KdV, and pure radiative models as defocusing NLS and Sinh-Gordon, while we discuss current limitations in other models such as focusing NLS (and Ablowitz-Ladik) and Sine-Gordon.
The Landau transformation xi = x/s(t) is the usual way to hold a moving boundary in place. It fails for fractional time derivatives because they have no chain rule: the transformed operator gains a nonlocal integral with a closed form only for self-similar fronts. We develop a sharp-interface finite-difference scheme for the conservation-consistent memory-flux fractional Stefan problem on a fixed grid, storing the memory limit h(x) = s^-1(x) as the time each node melts. The front is advanced using an integrated energy balance, with the singular source timed to the sub-step crossing instant. Against a known self-similar solution, the front constant is accurate to 0.02 to 0.11%, whereas competing formulations differ by 17 to 27% at alpha = 0.6. A remaining positivity violation follows from dt/t0 ~ s0^(1 - 2/alpha) and disappears on a front-locked time grid. We also derive two exact solutions with non-power-law fronts: a source-dominated solution and a Mittag-Leffler traveling wave, and verify the scheme against both. Neural-network solvers for moving boundaries are often trained in normalized coordinates; for alpha < 1, this silently changes the problem, while the change vanishes at alpha = 1, where checks are commonly performed. In the classical limit, our scheme is more accurate in 12 ms than a reimplemented moving-boundary network in 177 s.
In this paper, we study Egorov's theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch's quantization. In this quantization, the remainder in Egorov's theorem for eigenfunctions is given explicitly by the second derivatives of the symbols along the weak stable foliation of the geodesic flow. Forward in time, we show that Egorov's theorem for eigenfunctions is valid to all logarithmic times, far beyond the Ehrenfest time T_E=|log h|/2, where h going to 0 is the semiclassical parameter. Backward in time, we introduce the Schrodinger group along the stable horocycles, which absorbs the exponentially growing part of the remainder. We show that Egorov's theorem for eigenfunctions is valid up to -T_E+T_h for any Th going to infinity with 0<=T_h<=T_E. Moreover, the limit of the remainder in Egorov's theorem at the Ehrenfest time -T_E is exactly the difference between the semiclassical measure and the Liouville measure, tested against the symbols propagated by the Schrodinger group. Consequently, Egorov's theorem for a sequence of eigenfunctions holds at -T_E (and in fact, at -T_E-T for any fixed T in R if and only if the sequence tends to be equidistributed in the phase space. Thus, the quantum unique ergodicity (QUE) conjecture is equivalent to this Egorov property holding for every eigenbasis. Furthermore, QUE holds with a polynomial rate if and only if Egorov's theorem holds at -T_E with a polynomial remainder. In this case, Egorov's theorem holds beyond -T_E with a polynomial remainder, and the stronger the rate is in QUE, the longer the time is in Egorov's theorem.
In this paper, we construct Feynman propagators on exponentially decaying perturbations of stationary spacetime with compact Cauchy surface. The propagators are defined as parametrices of the wave operator on exponentially weighted Sobolev spaces defined by complex contours $\Gamma$. We then apply this theory to the computation of indexes of Dirac operators using a continuous Wick-like rotation to relate it to the Riemannian case.
We prove local existence and uniqueness, propagation of Sobolev regularity, and continuation criteria results for the characteristic initial value problem for the 3D compressible Euler equations. This is the Cauchy problem with initial data given on a pair of transversally intersecting (acoustically) null hypersurfaces. The region of existence in our main theorem is a future-neighborhood of the union of the data null hypersurfaces, where the size of the neighborhood depends on various norms of the data as well as the structure of the data null hypersurfaces. We allow for any equation of state with positive sound speed, and we do not make any symmetry, irrotationality, or isentropicity assumptions. Our analysis relies on the geometric formulation of the flow derived in [Speck19], which decomposes the equations into a "wave-part" coupled to a "transport-div-curl" part. Compared to Rendall's celebrated solution of the characteristic initial value problem for wave equations [Rendall90], substantial new ideas are needed to handle the fact that the compressible Euler equations are more complicated than pure wave equations.
In this article we establish new obstructions to the categorification of near-group fusion rings by combining the values of irreducible characters of the Grothendieck ring with the Galois symmetry of the induced unit object $I(1)$ in the Drinfeld center. As a result, we show that a near-group ring of type $G+m$, where $G$ is an abelian group and both $|G|$ and $m$ are odd, can only categorify if $m=|G|$.
In this note, we begin with considering the simple one-dimensional field theory describing a single free particle on manifold, and apply the BRST formalism and BV quantization to this model following the standard procedures in physics. Through calculations that, while not fully rigorous, have a more transparent physical interpretation, then subsequent geometric identifications, we recover the deformation quantization and algebraic index theorem on symplectic manifold. This note thus elucidates the physical origins and mechanisms of these mathematical results.
We consider a two-well problem within geometrically linear elasticity, on a bounded Lipschitz set in dimension $n\ge 2$. We assume that the two wells are compatible through a symmetrized rank-one connection and that the displacement matches their average on the boundary, and we include a regularization based on the total surface energy. We show that there are two distinct regimes, with the energy scaling differently in terms of the surface-energy coefficient $\eps$: it is proportional to $\eps^{2/3}$ if there are two distinct compatible twin-normal directions, and to $\eps^{4/5}$ if there is only one. This matches previous findings in rectangles and other special geometries, mostly in two dimensions, and the lower bounds obtained by a Fourier method by Rai\c{t}\u{a}, R\"uland, Tissot and Tribuzio (2024). Our results are obtained by explicit constructions, which contain a self-similar refinement of oscillations close to the boundary, and by a short real-space proof of the matching lower bounds.
We establish a conditional quantum Kac-Bernstein theorem for mixing of two independent, finite-mode bosonic inputs with quantum memories. We characterize exactly when the outputs are conditionally independent given both memories. Each input then decomposes into orthogonal memory sectors sharing a common Gaussian state on the physical system and a bosonic memory factor, up to sector-dependent physical displacements. In particular, triviality of the common bosonic factor implies separability of the inputs, as occurs whenever either memory has finite-dimensional support.
We study the self-consistent common-noise equation associated with the Weil--Petersson response differential of the rotationally marked SLE welding probability. For $0<\kappa\leq4$, we prove joint weak uniqueness among continuous adapted weak solutions whose law at each deterministic time is the welding probability, and obtain the Markov property in the original filtration. A conditional Gaussian lift and capacity localization give the required $L^1$ generator-range density. Identification with the canonical law gives stationarity and reversibility. We also construct autonomous Fej\'er and Poisson regularizations with pathwise-unique global strong solutions. Their joint state-noise laws converge to the canonical law for every initial probability absolutely continuous with respect to the welding probability.
In the context of heat conduction, the equations of Maxwell-Cattaneo is a successful alternative to the Fourier's law. In 2009, Christov proposed a frame-independent (or objective) generalization of Maxwell-Cattaneo equation. For a more general class of objective heat conduction Maxwell-Cattaneo-Christov equations proposed by Angeles, a uniqueness result is proved, applied to an incompressible fluid.
The aim of this article is to derive a stochastic interpretation of the Water Waves equations, based on the location uncertainty principle (LU). Next, these equations are used to infer simpler models from scaling limits, such as the Saint-Venant, Boussinesq and Serre-Green-Naghdi ones. Once adapted to the stochastic setting, these models have been successfully implemented numerically, and we show several experiments to assess their properties -- for example the breaking of the wave symmetry, and the influence of time correlated noises.
We present a geometric derivation of the hidden symmetries of the Kepler problem, combining a duality for central-force systems, obtained by interchanging positions and velocities, with a generalized form of Jacobi's theorem. Applied to the dual Kepler problem, this theorem identifies Kepler hodographs of prescribed energy with geodesics of a metric of constant curvature. After adjoining one point at infinity, the resulting geometry is complete. Its Killing vector fields induce a finite-dimensional Lie algebra of infinitesimal symmetries on the corresponding energy hypersurface. The distinguished point at infinity determines a canonical decomposition of the Killing algebra into two vector subspaces, whose associated Noether invariants recover angular momentum and the Runge-Lenz vector.
In this work, we prove exponential decay for the spin correlation of the spin $O(\mathsf{N})$ model for $\mathsf{N}\geq3$ at every inverse temperature $ \beta>0 $. In particular, our work confirms Polyakov's prediction in 1975 on the Heisenberg model, i.e., the case of $\mathsf{N}=3$. To this end, we investigate a generalized $O(\mathsf{N})$ model where $ \mathsf{N} $ is a positive \emph{real number}, as follows: we sample a loop soup on the metric graph (say of a box) with intensity $ \tfrac{\mathsf{N}}{2} $ subject to the conditioning that the local time at each lattice vertex is $\beta$, and then for each loop we independently assign it to copies $1\leq \mathsf{i}\leq \lceil\mathsf{N}\rceil$ such that it is assigned to copy $\mathsf{i}$ with probability $ \tfrac{1}{\mathsf{N}} $ for $ 1\leq \mathsf{i}\leq \lfloor\mathsf{N}\rfloor $. Then, by Lupu's isomorphism theorem, we can construct a spin field by assigning a random sign to each loop soup cluster in each copy independently, such that the field coincides with the spin $O(\mathsf{N})$ model when $\mathsf{N}$ is an integer. We then establish exponential decay for every real-valued $\mathsf{N}>2$. Our result reveals an underlying mechanism for exponential decay, which emerges as soon as $\mathsf{N}>2$ (i.e., as soon as beyond the range of the XY model) and does not necessarily rely on the rotation symmetry in the sphere $ \mathbb{S}^{2} $.
Twilight clouds are small, optically faint warm clouds whose droplets fluctuate between haze and activated growth. We develop a nonequilibrium statistical field theory linking this regime to cloud-field organization through K\"ohler microphysics, supersaturation transport, and turbulent mixing. At parcel level, supersaturation rapidly tracks slow droplet growth with a finite lag. Mixing-induced supersaturation removal drives a saddle-node bifurcation, shifting the activation threshold and supporting intermittent hovering under turbulent fluctuations. Starting from turbulent transport of supersaturation coupled to droplet growth, we derive a droplet-supersaturation field system and formulate it within the Martin-Siggia-Rose framework. A Gurari-type variational reduction inspired by polaron theory eliminates the fast bath. Turbulence dresses the slow droplet field, shrinks the forbidden zone below the equilibrium curve, and raises the mixing threshold required to suppress activation. Each droplet population induces a screened supersaturation halo that couples neighboring cloud elements; its delayed adjustment generates dynamic inertia and temporally correlated noise. The resulting nonlinear, non-Markovian pseudoparabolic stochastic equation has a bounded frozen-background response at short wavelengths. Its long-wave coupling amplifies contrasts where the dressed slow manifold turns downward. Comparisons with the parent two-field system show that a spatially correlated Markovian closure preserves droplet-radius distributions and cloudlet organization. Near the activation fold, restoring residual-bath fluctuation memory improves recurrent-hovering frequency and patch extent without recalibrating the drift or fitting a correlation time. More broadly, the framework identifies how slow fields acquire interactions and inertia through the relaxing, diffusive environments they modify.
We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for $\bigl(\zeta'/\zeta\bigr)^{k}$ on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function $\Lambda^{*k}$ in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of $\Lambda^{*k}$ over monic polynomials in $\mathbb F_q[T]$; we denote its leading coefficient by $\delta_k(c)$, where the total trace degree and matrix size grow with a fixed ratio $c$. Under a weaker shifted moments conjecture and RH, we recover this $\delta_k(c)$ as the leading coefficient of the variance of short-interval sums of $\Lambda^{*k}$ over the integers. It is the counterpart of $\gamma_k(c)$, the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of $\zeta(s)^k$ replaced by those of $\bigl(-\zeta'/\zeta\bigr)^k$.
In our previous article CMP 407 (2026) 131, we explored the rich algebraic structures arising from a specific quantum many-body Hamiltonian quartic in the bosonic creation-annihilation operators. This Hamiltonian originated from the works of G\'erard and Grellier on classical turbulent energy transfer in a Lax-integrable setting, and its quantum version respects hierarchies of polynomial conservation laws all of which possess purely integer spectra, as does the Hamiltonian itself. The latter feature, known from numerical studies, has previously remained unproved. Here, we present an elementary solution of the quantum G\'erard-Grellier Hamiltonian. The eigenstates are labelled by partitions and the corresponding eigenvalues admit simple explicit expressions in terms of the principal hook lengths. The key ingredient in this construction is a specific basis we introduce, inspired by the classical multisoliton coherent states, in which the Hamiltonian is upper-triangular. The expressions for the eigenvalues then immediately follow, while the full eigenbasis can be recovered via a Gram-Schmidt procedure.
We determine the monochromatic three-arm exponent for critical planar percolation through the exact spectral threshold of an explicit positive integral operator family with elementary kernels, obtaining $\xi_3=0.76928375\cdots$. Unlike the monochromatic two-arm (backbone) event, the three-arm event can be obstructed by two distinct white clusters, so a Markovian exploration must retain the past connections. We introduce a two-point boundary observable that records these connections. Using the coupling between conformal loop ensemble (CLE) and Liouville quantum gravity (LQG), this observable satisfies a renewal equation in terms of quantum boundary lengths, and yields the explicit operator whose spectral information determines $\xi_3$. Our approach also gives an explicit spectral characterization of monochromatic $n$-arm exponents for all $n\ge4$.
Let $\mu$ be a probability distribution on a spin system with $n$ sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix $A$ with zero diagonal and $\lambda_{\max}(A)<1$ encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance $(I-A)^{-1}$ attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the $O(N)$ model, with $N\geq 2$, arbitrary external fields, and couplings of either sign, has spectral gap at least $(1-\|J_{\mathrm{abs}}\|_{\mathrm{op}}/N)/n$, where $J$ is the interaction matrix and $J_{\mathrm{abs}}=(|J_{ij}|)$. We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over $\mathbb F_q$, we obtain a one-sided local spectral expansion bound of $2\sqrt q/(q+1)$. This bound is independent of the number and sizes of each group, and is nontrivial for every prime power $q$. Finally, for random forests $F$ of any finite graph with arbitrary positive edge weights, we prove that \[ \Pr(e,f\in F)\le \frac{4}{3} \Pr(e\in F)\Pr(f\in F) \] for distinct edges $e,f$, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.
Minimal model and its linear stability analysis (LSA) for Plateau-Rayleigh (PR) instability of an intermixed, solid binary alloy nanowire is presented. Model accounts only for the surface diffusions of the alloy components without anisotropies of a surface energies, and for a classical thermodynamic surface segregation. Model allows to assess a deviation of a longwave instability cutoff wavelength $\lambda_c$ from the classical value $2\pi r_0$ (a wire circumference) as a function of a surface composition deviation from its equilibrium value, a wire radius, and a bulk concentration of a minority component. Results of LSA are presented for AuNi, PtNi, and CoNi nanowires, where an important distinction is that a surface segregation curve for CoNi is qualitatively different from the one for AuNi and PtNi. For each nanowire there is a sizable region of a parameter space where $\lambda_c<2\pi r_0$ and correspondingly, a fastest growing perturbation wavelength $\lambda_{max}$ is also less than a wire circumference. This is in stark contrast to PR instability of a single-component nanowire transporting mass by surface diffusion - in this case $\lambda_c$ and $\lambda_{max}$ are larger than a wire circumference. Strong surface segregation is primarily responsible for unusually small $\lambda_c$ and $\lambda_{max}$. Results suggest that a large number of small and closely spaced nanoparticles may emerge upon disintegration via annealing of a binary alloy nanowire.
We study the hard-core variant of the Simplified Approach to the ground state of the dilute Bose gas. The hard-core potential has been the subject of renewed attention, and is physically relevant to realistic atomic Bose gases. The hard-core Simple Equation is an integro-differential equation, and we construct its solution via a monotone iteration scheme and prove existence and uniqueness (in a restricted sense) of the solution. The energy per particle $e$ and the density $\rho$ of the solution are shown to satisfy the Lee-Huang-Yang formula $e = 2\pi\rho(1 + \frac{128}{15\sqrt{\pi}}\sqrt{\rho} + o(\sqrt{\rho}))$ as $\rho \to 0$, and the condensate fraction $\eta$ predicted by the hard-core Simple Equation is shown to satisfy $\eta = 1 - \frac{8}{3\sqrt{\pi}}\sqrt{\rho} + o(\sqrt{\rho})$ as $\rho \to 0$, in agreement with Bogolyubov's theory. We thus extend results of Carlen, Jauslin, Lieb to the hard-core case. The hard-core equation is defined on its own terms, and the proofs proceed by approximating the hard-core potential with the soft-core potentials $m\,\chi_{\{|x| \le 1\}}$ of increasing height $m$: at fixed energy, we establish the $L^1$ convergence of the corresponding solutions and the convergence of their densities at an $O(m^{-1/2})$ rate, and we show that the Lee-Huang-Yang expansions of the energy and of the condensate fraction for the soft-core equations hold with errors uniform in $m$, so that they survive the hard-core limit.
We construct an exact map between continuum fermion fields and lattice fields in 1+1 dimensions. This map rewrites a continuum Hamiltonian as a lattice Hamiltonian with nearest-neighbor couplings and an infinite-dimensional local Hilbert space. The resulting lattice model reproduces exactly the energy spectrum, global operator algebra, chiral symmetries, and anomalies of the continuum QFT on any finite lattice. Unlike domain wall fermions, truncating the local Hilbert space introduces no extra massless modes, and the entanglement entropy of an interval converges as the truncation is removed. The lattice local algebras are of type I and lie between the type III continuum algebras of nested spatial regions, in accordance with the split property. We demonstrate this construction explicitly for $N$ Majorana fermions, realizing the $\mathrm{O}(N)_\mathrm{L}\times\mathrm{O}(N)_\mathrm{R}$ lattice chiral symmetry. We also construct an exact lattice model for the Schwinger model and numerically reproduce the mass gap $e/\sqrt{\pi}$.
Harrop-Griffiths, Killip, Ntekoume and Vi\c{s}an proved that the derivative nonlinear Schr\"odinger equation (DNLS) is globally well-posed in $L^2(\mathbb{R})$. A key ingredient of their proof is a local smoothing estimate. It holds on $L^2$-equicontinuous sets of initial data, but it cannot hold on $L^2$-balls, because rescaled stationary solitons, all of mass $2\pi$, violate it. This raises a natural question: for which masses $m$ does local smoothing hold uniformly on the ball $\{M(q_0)\le m\}$? We prove three results. First, we give a quantitative version of the local smoothing estimate in which the data enter only through the mass and an explicit frequency scale $\kappa(q_0)$. On time intervals of length $2T$ the bound grows like $1+\kappa(q_0)^2T$, and the stationary soliton shows that the power of $\kappa$ cannot be lowered below one. Second, we prove uniform local smoothing on balls of small mass, together with a scale-invariant Morrey-type form and a localized $L^4_{t,x}$ bound. Third, we analyse the soliton family from both sides. The uniform constant must blow up at least like $(2\pi-m)^{-1}$ as $m\uparrow 2\pi$, while every soliton of mass $m<2\pi$ satisfies local smoothing with a constant $O((2\pi-m)^{-1})$. These facts single out the stationary-soliton mass $2\pi$, which is also the Hayashi--Ozawa coercivity threshold, as the natural candidate for the sharp threshold. We formulate this as a conjecture, give a spectral reformulation in terms of the Kaup--Newell scattering data, and discuss the obstacles to a proof.
We consider the sign clusters of the Gaussian free field on the cable system of a transient weighted graph with volume growth exponent $\alpha$ and Green function decay exponent $\nu$. We prove that the probability that the cluster of a vertex reaches distance $R$ is of order $R^{-\beta}$ when $2\alpha<3\nu$, where $\beta=\alpha-\nu$ is the walk dimension, and that the same holds up to a logarithmic factor in the upper bound when $2\alpha=3\nu$. On $\mathbb Z^d$ these conditions read $d>6$ and $d=6$. The upper bound combines a general entropic bound, which compares the law of the cluster under the critical loop soup and under a massive one, with a Simon-Lieb-type inequality for the massive field derived from its Markov property. The arguments are robust and require no transitivity. The lower bound follows from a competition between the capacity and the volume of the cluster. In both bounds, the key quantity is the expected exit time of the random walk from balls, which links the volume of the cluster to its radius.
We construct boundary Liouville conformal field theory at central charge $c\leq1$ with FZZT-type boundary conditions and show that it satisfies the Cardy--Lewellen sewing constraints on the disk and the annulus. Using the analytic bootstrap we derive the so far missing boundary three-point and bulk-to-boundary structure constants and a consistent disk one-point function as explicit meromorphic functions of the Liouville momenta. In sharp contrast to $c\geq25$, the open-string spectrum between two ``timelike'' FZZT boundaries is \emph{discrete}, yet non-trivially consistent through open-closed duality with the \emph{continuous} closed-string spectrum of Ribault and Santachiara.