Papers

When the control field of a qubit, a polarization state, or a spin-$\tfrac12$ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle $\Omega[C]$ intrinsically, and displacing the degeneracy by $\epsilon\uhat$ closes the path with enclosed solid angle $\Omega(\epsilon\uhat)=\Omega[C]+2\alpha+O(\epsilon)$, where $\alpha$ is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic—the great circle in the osculating plane ($\alpha=0$)—supplied by the curvature at the degeneracy. Berry's $\pi$ invariant under reversal of the displacement and the values $\pm\pi/2$ under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.

Reading an optical quantum reservoir costs repetitions growing super-exponentially with feature order: what it can afford is set by its detector, not its optics. For reservoirs encoding data in a parametric pump's phase, one conservation law fixes what is readable and what it costs. Pairwise photon exchange conserves an integer phase charge: across an ensemble of input masks, order-D readout reaches exactly the assemblies of at most D unit-charge kernels; degree-one homodyne readout is universal for fading-memory functionals along weak-squeezing families, at shot cost polynomial in accuracy; and at fixed squeezing no finite degree reaches every sector, at a computable distance. Nonlinearity sits in the optics, not the detector, whose per-shot variance is fixed at every order. In a hardware-faithful digital twin-no device was built-the rule is visible: on an open RF corpus a displacement-encoded control at identical photon number loses 17.7 accuracy points, as the charge algebra predicts.

We revisit the Moore–Read construction of the $\nu=1/k$ Laughlin states on compact Riemann surfaces, deriving their conformal blocks from $U(1)_k$ Chern–Simons theory through the CS/WZW correspondence. We compute their explicit monodromies under quasi-hole transport and Aharonov–Bohm flux insertion, and conjecture that these coincide with the corresponding adiabatic holonomies. Under this identification, we recover classical results on Laughlin states including the flux-averaged Hall conductance $1/k$ via a vector bundle of Laughlin states over $Jac(\Sigma)$. We also relate our construction to the recently developed algebro-geometric approach to higher genus Laughlin states.

We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.

We construct a semistrict higher Chern–Simons (HCS) gauge theory in $2n+2$ dimensions associated with balanced 2-term $L_\infty$-algebras. Starting from the homotopy Maurer–Cartan theory, we first introduce 2-term $L_\infty$-algebra gauge theory, and show that there is a four-dimensional HCS construction. Then we extend invariant bilinear pairings to invariant multilinear forms of the appropriate degree, and define a $(2n+3)$-dimensional higher Pontryagin–Chern form, which is closed and invariant under infinitesimal gauge transformations. Its transgression yields an explicit $(2n+2)$-dimensional HCS form. We further establish a higher Chern–Weil theorem that generates higher transgression forms, and prove that the HCS theory is a distinguished instance of the higher transgression gauge theory. Finally, we apply the extended Cartan homotopy formula in this semistrict setting, and show that it is a common origin of both the higher Chern–Weil theorem and the associated triangle equation.

It is well established that in time-dependent quantum systems, integrability preserving time-dependent interaction strengths are identical to the renormalization group (RG) trajectories of the corresponding static model when time `$t$' in the driven model is identified with the logarithm of the cutoff `$\log\Lambda$' of the static model. We refer to this integrability preserving driving as the RG protocol. In this work we extend the class of time-dependent integrable models to include non-Hermitian quantum models with time-dependent interaction strengths. Using the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that the constraints imposed by integrability are more general: The interaction strengths of the static model that flow in the RG follow the respective RG trajectories in the corresponding time-dependent model as described above. In addition, the interaction strengths of the static model that are RG invariant can either be constant or have a specific time-dependence in the corresponding time-dependent model which is constrained by integrability. Thus we establish that in the context of time-dependent non-Hermitian systems, the set of integrability preserving time-dependent strengths is larger than the set corresponding to the RG protocol.

Simulating dispersive shock waves (DSWs) using physics-informed neural networks (PINNs) presents significant numerical difficulties. In particular, the square-root initial condition exhibits singularity near the origin, making it difficult to fit accurately. This initial fitting error is amplified through nonlinear evolution and gradually accumulates during extrapolation, imposing high demands on the long-term extrapolation capability of numerical methods. Accordingly, this paper proposes an improved physics-guided multistage neural network (IPgMSNN) framework. IPgMSNN is built upon the original PgMSNN framework, with its core innovation focused on the third-stage training strategy. Specifically, online fine-tuning and a weighted target mechanism are introduced, which effectively suppress error accumulation during long-term extrapolation. Numerical experiments systematically evaluate IPgMSNN from three perspectives: forward problem solving, model stability, and parameter inversion. Experimental results demonstrate that IPgMSNN effectively captures high-frequency wavefront details, maintains long-term extrapolation stability, and achieves high-precision parameter inversion, providing an efficient solution for deep learning-based simulation of dispersive nonlinear systems.

We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$, and states of mean energy at most $E$, the supremum of the modular energy is $M_{\max} = \frac{\pi R}{2}E + \frac{c}{6} - \frac{\pi c^{2}}{288\,RE} + O((RE)^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\min} \simeq -\frac{c}{12}[\ln(24\pi RE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE \to 0$; it is strictly wider above, by more than $\frac{2}{5}\pi RE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $\pi R/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $\Delta S_I \leq \frac{\pi R}{2}E + \frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.

Even for tension-dominated fracture propagation, porous materials may deform plastically adjacent to the propagating fracture. As is common for porous materials, existing phase-field models typically employ a non-associative flow rule for plasticity, and a Helmholtz free energy based on strain and fluid pressure. This work revisits the hydromechanically coupled formulation of the phase-field model for fracture in poro-elastoplastic media by analyzing the strength surface and fracture driving force. Our analyses show that these common choices of flow rule and free energy will lead to a discontinuous strength surface across the tension-compression transition. A non-associative flow rule introduces a jump at the strength surface, while treating fluid pressure-rather than fluid content-as the independent variable in the Helmholtz free energy omits a coupling term from the phase-field driving force, also breaking continuity. Incorporating an associative Drucker-Prager flow rule and this omitted coupling term ensures a continuous strength surface and the accurate fracture driving force. The proposed model exhibits improved accuracy in hydromechanical responses when compared against the analytical solution of the Kristianovich-Geertsma-de Klerk hydraulic fracturing benchmark. Numerical simulations of hydraulic fracturing and biaxial compression in poro-elastoplastic media show that the model can reproduce both shear-dominated fractures induced by mechanical disturbance and tension-dominated fractures driven by fluid injection in saturated porous media.

We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^2((0,\infty);\sqrt \omega \dd\omega )$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.

At fixed pulse energy, which drive waveform extracts the most $n$th-order nonlinear conversion from a resonator ($n=2$ for second harmonic)? A short pulse couples poorly to a narrow resonance, a long one dilutes its energy, and no linear rule fixes the compromise. We solve the problem exactly for a single mode of amplitude decay rate $\kappa$. Eliminating the drive turns fixed incident energy into a constraint on the stored field alone, and the optimization becomes a sharp Gagliardo–Nirenberg inequality whose extremal is the ground-state soliton of the nonlinear Schrödinger equation. The optimal stored field is $\mathrm{sech}^{1/(n-1)}[(n-1)\kappa t]$, sustained by an asymmetric input that rises as $e^{\kappa t}$ and falls as $e^{-(2n-1)\kappa t}$; the largest converted energy follows in closed form. A rising exponential, the time-reversal recipe, retains at most $79.0\%$ of the bound at $n=2$ and $2/e$ at large $n$; a two-rate pulse retains above $97\%$. Critical coupling generalizes to $n$-fold overcoupling, with optimal input coupling $n$ times the intrinsic loss rate. The bound applies from microrings to superconducting circuits and caps the per-pulse brightness of broadband photon-pair sources.

Partial entanglement may provide enough connections in the universe to satisfy the definition of a small world. To investigate this possibility, we define a network of particles on a given space-like hyper-surface with a long-range link between any two particles that are connected by a chain of exchanged particles involving a limited number of interactions. Considering the mean free paths of particles in different regions of space and the resulting probability distributions of entanglement connections vs. distance, we find evidence of small-world or random network structure on all but the smallest scales - corresponding to stars and planets - on which other types of connections would need to be added to complete the small world picture.

We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$. By analyzing the convergence of the perturbative expansion in the interaction strength $\lambda$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $\lambda |t|^{2D+1}=\mathcal{O}(1)$, when the Hamiltonian is geometrically local on a $D$-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $\lambda |t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $\lambda |t|=\mathcal{O}(1)$, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.

These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill–Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.

We establish an optimal Poincaré inequality for real-valued symmetric observables of the complex Ginibre log-gas. Equality is attained by the real and imaginary parts of the center-of-mass observable. Equivalently, we determine the exact spectral gap of the associated overdamped Langevin dynamics, for real symmetric observables. The Hessian of the energy of this log-gas is unbounded below, so standard convexity arguments do not directly apply. The proof instead combines a Vandermonde transform, a holomorphic projection, and a complex Gaussian d-bar spectral-gap estimate, corresponding to the constant-curvature case of the Hörmander-Berndtsson estimate. It is short and self-contained. It remains valid, beyond the quadratic potential, for rotationally invariant additive plurisubharmonic potential perturbations. Additionally, we provide four alternative proofs, two of which yield sum-of-squares formulas for the deficit, based respectively on a Hermite expansion and on an integrated Bochner-Kodaira formula, the other two use the spectral analysis of a two-sided number-operator factorization and a Hermite-Slater polynomial expansion. We furthermore present new results related to linear statistics, Gaussian factorization, polynomial eigenfunctions, log-Sobolev inequalities, matrix lift and eigenvector overlaps, and non-quadratic potentials.

The connective constant $\mu(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_\omega$ of a general Grigorchuk group encoded by a sequence $\omega\in\{0,1,2\}^{\Bbb N}$. In particular, $\mu(G_\omega) > \phi$ for any such Cayley graph (subject to a simple condition on $\omega$), where $\phi:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean”, Discrete Math. 343 (2020), article 111638, where it was conjectured that $\mu(G)\ge\phi$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.

We study a long-range perturbation of the perimeter functional by a nonlocal repulsive term defined through the Yukawa kernel, minimized under a volume constraint. Because the kernel decays exponentially, the competition between surface tension and repulsion is governed by two independent quantities, the screening rate and the volume. Specifically we prove that (i) above an explicit critical screening rate, minimizers exist at every volume, whereas earlier work required the volume to be large; (ii) below an explicit volume threshold minimizers exist for every screening rate; (iii) at small volume, and uniformly in the screening rate, the ball is the unique minimizer up to translation; and (iv) there exists a sharp volume threshold, depending on the screening rate, for the ball to be a stable volume-constrained critical point. The threshold is given in closed form for every screening rate and reduces to the known unscreened value when $\alpha=0$. Our proofs overcome new technical changes due to the lack of homogeneity in the Yukawa kernel.

We define a map from solutions of the Bethe Ansatz equations (BAEs) of four-dimensional $\mathcal{N}=4$ super-Yang–Mills with arbitrary semisimple gauge algebra $\mathfrak{g}$ to extrema and poles of the potential of the untwisted elliptic Calogero–Moser system of type $\mathfrak{g}$. We conjecture the map to be a bijection on the preimage of the Calogero–Moser extrema, and show that it intertwines the symmetries of the two systems, both the gauge ones (torus, Weyl and center invariance) and a $\mathrm{PSL}(2,\mathbb{Z})$ action, so that solutions on both sides organize into orbits, each BAE orbit mapping onto a single orbit of Calogero–Moser extrema or poles. That the system produced is the untwisted one has a consequence: the conjectured correspondence between BAE solutions and vacua of the $\mathcal{N}=1^\ast$ deformation of $\mathcal{N}=4$ on $\mathbb{R}^{3,1}$, which are extrema of the twisted system, cannot extend to non-simply-laced $\mathfrak{g}$. It also fails within the simply-laced cases, though not for $\mathfrak{su}(N)$: we exhibit an $\mathfrak{so}(8)$ solution that flows to a pole of the Calogero–Moser potential rather than to an extremum, and so has no $\mathcal{N}=1^\ast$ counterpart. We illustrate the map in detail for every rank-two $\mathfrak{g}$, classical and exceptional alike, and use these cases as evidence for the conjecture.

Quantum state transfer on qubit lattices is a crucial step in a panoply of quantum information processing tasks. Understanding its fundamental limits in the presence of practical imperfections remains a pressing open question. In particular, it is unclear how thermal noise in the intermediate ancilla sites impedes transfer. In this work, we derive tight lower bounds on the necessary growth of commutator norms to achieve approximate state transfer. Specializing to 1D power-law systems with thermal ancilla states, we demonstrate that state transfer runtimes depend sensitively on the scaling of temperature with system size, improving logarithmic bounds to algebraic ones. Our work extends previous results on exact state transfer to the qualitatively different and more practically relevant regime of approximate state transfer.

Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on $M$ optical modes, with $K$ equally squeezed input modes and $N$ observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers $K$, which is a part of the argument for classical hardness of GBS. In particular, we show that for any $K$ and $N=o(\sqrt{K})$, the symmetric product $MK^{-1/2}U_{NK}U_{NK}^T$, for $U_{NK}$ the top left $N\times K$ submatrix of an $M\times M$ Haar random unitary $U$, is close in total variation distance to both an $N\times N$ symmetric complex Gaussian matrix $\mathbf G$ with independent entries, and the symmetric product $GG^T/\sqrt{K}$ for $G$ an $N\times K$ matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with $K=cM$ if $c<1/2$. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.

In view of embedding 3D gravity into topological string theory, we show that the dimensional reduction of 6D Kodaira-Spencer gravity on $\text{AdS}_3\times S^3$, i.e. the low-energy description of twisted holography, contains a subsector that coincides with the action of chiral 3D gravity on $\text{AdS}_3$. Furthermore, we show that the full 3D Einstein action is obtained as the reduction of two independent complex-conjugate copies of 6D Kodaira-Spencer gravity. We perform the dimensional reduction explicitly at the level of the classical actions, extending our previous work that related the equations of motion of the two theories. Our reduction procedure is based on a novel rewriting of the non-local 6D Kodaira-Spencer action as a local 6D holomorphic Chern-Simons action for the gauge group $\mathrm{SL}(2,\mathbb{C})$, valid within a subsector of complex structure deformations that we identify. Our results provide a necessary step toward embedding the Euclidean path integral of 3D gravity into topological string theory.

We study the long range behavior of solutions to $i\partial_tu=H_\alpha u+\lambda|u|u$ on $\mathbb R^2$, where $H_\alpha$ is the Friedrichs realization of the Aharonov-Bohm Hamiltonian with a single pole. The logarithmic phase of the long range ansatz may push a profile out of the domain of $H_{\alpha}$. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order $\le \frac 12$; at half flux $\alpha=\frac 12$, no nonzero trace survives. However, every profile in the full domain of $H_{\alpha}$ with small $L^\infty$ amplitude determines a unique global solution with a modified final state, with a remainder rate $t^{-b}$ for all $0<b<1/2+\nu_\alpha$, $\nu_\alpha=\min\{\alpha,1-\alpha\}$. For profiles satisfying the vanishing trace condition, the rate improves to every $0<b<1$. This result is sharp in the sense that, if $\alpha\neq \frac 12$, we can construct profiles with an error of size $t^{-1/2-\nu_\alpha}\log t$, ruling out all faster rates. The upper bound comes from a retarded Strichartz estimate for a residual that is not in $L^2$; the lower bound is an explicit calculation via Hankel transforms. For smoother profiles we also compute the first correction, which gives remainder rates with $1<b<2$.

The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang–Kueng–Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor $(3/2)^k$ for $k$-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within $2/(d+2)$ of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains $(3/2)^k$. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.

Representations of the Kohno-Drinfeld Lie algebra associated with several classical multivariate distributions have played an important role in recent developments in the theory of quantum superintegrable systems. In this work, we analyze the corresponding Gaudin models for the multivariate Hahn, Dirichlet, multinomial, and negative multinomial distributions. More precisely, using tools from representation theory, we construct multivariate orthogonal polynomials with respect to these distributions as common eigenfunctions of Gaudin operators. The polynomials are parametrized by solutions to the Bethe ansatz equations or, equivalently, by the roots of Heine-Stieltjes polynomials.

Many rigorous results in the modern era of statistical mechanics have been obtained through geometrical representations. A much studied tool in this setting is the random cluster representation of lattice spin models. Another area of statistical mechanics that is of great interest but lacking rigorous results is the theory of the renormalization group. This paper investigates the idea to find a common ground between these two concepts in order to obtain rigorous results on the renormalization group flow. A need for negative cluster-weights weakens the success of this approach. Nevertheless, we managed to establish a relation between the scaling limit of the renormalization group flow of the nearest-neighbour Ising model away from criticality with a simple one-dimensional dynamical system that follows the cluster connectivity of the renormalization group transformation. This allows for a rigorous proof of the convergence of this flow to the zero- and infinite-temperature fixed point respectively for a large family of renormalization group transformations. Explicit results will be established on $\mathbb{Z}^2$ followed by a discussion of the available generalizations to higher dimensions.

Let $D\subset\mathbb R^d$ be a bounded domain, let $\kappa>0$ be fixed, and let $W$ be a fractional Brownian sheet on $\mathbb R\times\mathbb R^d$. Consider the Stratonovich parabolic Anderson model (PAM) $\partial_tu_\kappa=(\frac12\Delta+\kappa W')u_\kappa$ with Dirichlet boundary condition on $D$ and the flat initial condition $u_\kappa(0,\cdot)=\mathbf 1_D$. We calculate exact asymptotics for the expectation and the standard deviation of the total mass $\int_Du_\kappa(t,x)~\mathrm d x$ as $t\to0$ under the assumption that $W$'s Hurst indices are all at least $1/2$ and that $u_\kappa$'s moments are finite for small enough $t>0$. In doing so, we uncover that these asymptotics are determined by a competition between three mechanisms: (1) $\mathbf{Geometry}$: The rate of heat diffusion through the boundary $\partial D$. (2) $\mathbf{Fluctuations}$: $W$'s time Hurst index. (3) $\mathbf{Renormalization}$: The singularity of deterministic Stratonovich corrections. As a result, we identify novel phase transition phenomena, which arise from the influence of $W$'s Hurst indices on the relative magnitudes of these contributions.

We extend the notion of Plancherel convergence, introduced in the study of compact quotients of locally compact groups, to finite-area hyperbolic surfaces and establish a quantum ergodicity theorem for Plancherel sequences. We show that the expectation values of integral operators whose kernels are uniformly bounded and have uniformly bounded propagation become asymptotically equidistributed, on average, for eigenvalues in a fixed compact interval. Our result recovers the case of multiplication operators studied by Le Masson and Sahlsten, corresponding to the degenerate case of propagation bound zero and kernels supported on the diagonal. Moreover, our result allows the systole to shrink without a uniform lower bound, with the allowed degeneration constrained by Plancherel convergence. This identifies Plancherel convergence as a natural spectral framework for quantum ergodicity on degenerating hyperbolic surfaces.

We study a family of the non-vacuum stationary axisymmetric type D metrics, where the two principal null directions are geodesic and shearfree. It is demonstrated that the conformal-to-Carter metric gives the most general form of this family, without any assumptions which are previously used by Ovcharenko and Podolský to arrive at this conclusion.

The periodic non-Hermitian Baxter-Fendley $Z_N$ clock chain has lacked a complete finite-size spectral solution, whereas its open-chain counterpart admits a solution in terms of independent quasienergies. For the periodic model we show that the operator-valued matching polynomial associated with its cyclic Weyl algebra simultaneously generates a set of conserved quantities, including the Hamiltonian, and realizes a cyclic $\tau^{(2)}$ Yang-Baxter transfer matrix. Root-of-unity closure yields a finite system of polynomial spectral equations in each charge sector, which reproduces the complete finite-size energy spectrum counted with algebraic multiplicity. As a first application of this result, we show that Newton continuation of these equations provides a practical numerical route to the periodic ground-state energy without enumerating the full spectrum. For homogeneous chains the thermodynamic seam response yields a criterion for boundary-induced criticality; for $N=3$ it predicts two reciprocal critical couplings with singular ground-state curvature, in contrast to the single self-dual open boundary critical point.

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlevé-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.