$Φ$-intermediate dimensions interpolate between Hausdorff and box-counting dimensions by restricting admissible coverings to scale windows of the form $[Φ(r),r]$. Using a family of $Φ$-dependent kernels, we develop a potential-theoretic framework that characterizes these dimensions in terms of capacities and leads to associated $Φ$-dimension profiles. This framework provides effective tools for obtaining lower bounds from uniform potential estimates. As an application, we prove Marstrand--Mattila type projection theorems, showing that for $γ_{n,m}$-almost all $m$-dimensional subspaces $V$, the $Φ$-intermediate dimensions of $π_V E$ coincide with deterministic profile values depending only on $E$ and $m$. We also discuss consequences for continuity at the Hausdorff end-point and for the box dimensions of typical projections.

This work presents an updated version of the previously released freeware MAN (Modal Analysis of Nanoresonators) [Comput Phys Commun 284, 108627 (2023)], a software package designed to efficiently compute and normalize quasinormal modes (QNMs) and to use them for the analysis of the optical response of electromagnetic resonators. The current release introduces three major enhancements. First, Version 9 extends the capabilities of Version 8 by incorporating new models and functions dedicated to systems involving two-dimensional materials, such as graphene. Second, based on a newly developed coupled-QNM theory, a new toolbox has been implemented that enables the computation of the complex coupling coefficients between the modes of coupled resonators using the individual QNMs of the uncoupled cavities. Finally, we introduce new functions that allow for the direct evaluation of Fano parameters for the extinction cross-section directly from the QNM field distribution.

The mean pairwise velocity of massive halos reflects the gravitational force law on cosmic scales. We combine cosmic microwave background intensity maps from the Atacama Cosmology Telescope and a galaxy catalog from the Sloan Digital Sky Survey to estimate the mean pairwise velocity using the kinematic Sunyaev-Zeldovich (kSZ) effect. On scales from 30 -- 230 megaparsecs, we constrain the gravitational acceleration between pairs of halos at separation $r$ to be $g\propto 1/r^n$ with $n=2.1\pm 0.3$, which is consistent with Newtonian gravity in an expanding spacetime (\textit{i.e.}, the standard $Λ$CDM model). This constraint shows agreement with an inverse quadratic radial dependence over the large distances separating galaxy halos, as expected in standard cosmology. Upcoming surveys have the potential to rule out $n = 1$ at $10σ$ significance. Our results establish the kSZ effect as a powerful tool for testing gravity on cosmological scales.

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

Boson sampling is a leading candidate for demonstrating quantum advantage in photonic systems. Despite significant experimental and theoretical progress, a characterization of its output statistics remains incomplete. This is especially true beyond the dilute regime, where photon collisions and bunching become significant. The associated saturated regime, characterized by mode number growing linearly with photon number, or more generally sub-quadratically, is precisely the regime of greatest experimental interest. As a consequence, anti-concentration of the output distribution--a key ingredient in hardness arguments--remains poorly understood in boson sampling. In this work, we leverage representation-theoretic tools to address this gap, obtaining closed-form expressions for second moments of generic particle-number-preserving bosonic observables. We express these quantities in terms of Hilbert-Schmidt norms of projections onto irreducible components of the operator space and show that these projection norms admit compact analytical expressions by exploiting the underlying symmetry structure. Focusing on Fock state output probabilities, we further establish anti-concentration beyond the dilute regime. Together with recent complexity-theoretic results, our findings strengthen hardness guarantees for boson sampling in experimentally interesting settings.

We present the implementation of next-to-next-to-leading order (NNLO) electroweak (EW) virtual corrections at next-to-leading logarithmic (NLL) accuracy in the amplitude generator OpenLoops. The implementation covers the automated computation of processes involving massless fermions and transversely polarised vector bosons. For energies above the EW scale, logarithmic EW corrections are strongly enhanced in the tails of kinematic distributions of key LHC processes, reaching several tens of percent at NLO and several percent at NNLO. The two-loop implementation is validated against analytical results from the literature. We present phenomenological results for representative LHC processes and discuss the role of two-loop EW corrections in reducing theoretical uncertainties from missing higher-order contributions.

We consider the interacting Bose gas in the thermodynamic limit in a large box in $\R^d$ at positive temperature $1/β\in(0,\infty)$ with particle density $\simρ\in(0,\infty)$. We follow a path-integral approach and adopt from \cite {ACK10} a description of the free energy in terms of the {\it Brownian loop soup}, a Poisson point process consisting of Brownian bridges, also called loops or cycles. It is the objective of this paper to derive, for any values of $β$ and $ρ$, a formula for the limiting free energy with explicit control on the particle numbers in the short and in the long loops. The latter are presumed to play the role of the condensate, according to Feynman's \cite{F53} famous, vague suggestion, and they turn into {\it random interlacements} (bi-infinite, locally finite random processes in $\R^d$) in our formula. In \cite{ACK10} there was no concept that could describe the long loops; only small $ρ$ could be handled successfully. In the present paper we represent the limiting free energy in terms of a variational formula, ranging over the set of all stationary point processes with loops and with interlacements, having each a given particle density, and minimizing the sum of the interaction energy and a characteristic entropy term. The latter is a new kind of a {\it specific relative entropy density} with respect to the reference process of loops (the Brownian loop soup), together with an independent Markov kernel describing collections of path shreds in large boxes. In $d\geq 3$, the latter can be seen as a projection of the {\em Brownian interlacement Poisson point process with $β$-spacing}. Our proof tool box comes from large-deviation theory, both for the derivation of the formula for the free energy and for the proof of the existence of the specific relative entropy.

High- to ultrahigh-redshift clustering of halos provides a powerful tool to understand cosmology and galaxy formation. However, theoretical predictions are not firmly established in the first billion years, where current and upcoming surveys are beginning to reach percent-level precision. Here we measure dark matter halo biases at $z=6$ - 19 from simulation data, and find they are $\sim$ 3 - 4$\%$ higher than canonical results calibrated at low $z$. We provide an updated linear-bias fit at these early times, reducing the mean systematic offset to $< 1\%$. These results will enable robust interpretation of early-Universe galaxy clustering from JWST, Roman, and intensity-mapping surveys.

Dynamic soaring allows seabirds to harvest mechanical energy from vertical wind shear, but field trajectories lack a benchmark for comparing flight performances across species. We derive a reduced lower bound on transport effort from a simplified Hamilton-Jacobi-Bellman optimal-control model in which slow flight incurs an induced-drag penalty, fast flight incurs a dissipative penalty, and wind shear supplies an effective energetic subsidy. After species-specific normalization of transport speed and an accelerometer-based effort proxy, we map wandering albatrosses, Cory's shearwaters, and Eurasian oystercatchers into a common reduced speed-effort plane and estimate their empirical lower frontiers. The albatross frontier lies closest to the reduced bound, consistent with near-optimal wind-energy harvesting. The shearwater frontier is systematically displaced above it, and oystercatchers occupy a distinct non-soaring regime. The resulting framework places specialist dynamic soaring, mixed flap-gliding, and non-soaring flight in a common mechanical representation and provides a reduced benchmark for comparing wind-assisted flight across species using field trajectories.

The combination of control barrier functions (CBFs) and adaptive control -- a framework referred to as adaptive safety -- has proven to be a powerful paradigm for safety-critical control of nonlinear systems with parametric uncertainties. Yet the theoretical conditions for forward invariance within this framework are often quite conservative, and may require using large adaptation gains to achieve acceptable performance, an approach that is traditionally discouraged in adaptive control. This paper mitigates these issues via high-order tuners, a recent class of higher-order adaptation laws that leverages different adaptation gains at different orders of differentiation. We illustrate that these high-order tuners decouple adaptation gain conditions from those placed on the initial conditions of the system required for set invariance. We extend these results to robotic systems whose linear-in-the-parameters structure proves particularly useful for adaptive control. The efficacy of our results are illustrated via simulations.

We develop a method for simulating colloidal suspensions using multiparticle collision dynamics (MPCD) with a discrete particle model represented as a rigid body. The key steps for incorporating the rigid-body constraints are to thermalize the velocities of the discrete sites before they participate in the MPCD collision step, then transfer momentum from the sites to the rigid body. We demonstrate that the rigid-body model produces the expected statistics for a single spherical particle and the same transport properties for a hard-sphere colloidal suspension as an equivalent model using harmonic bonds to maintain the site geometry. Importantly, the rigid-body model has less computational overhead and permits a larger simulation timestep than the harmonic-bond model, leading to a nearly order of magnitude speedup in benchmark simulations of hard-sphere colloidal suspensions. Our method is compatible with arbitrary discretization, so it enables more efficient MPCD simulations of suspensions of colloidal particles with complex shapes.

We explore the mechanics of a terminally loaded buckled elastica under frictionless self-contact. With the aid of two integrals associated with the elastica, we propose a scale-invariant condition necessary for the onset of contact. The condition is independent of the boundary conditions, does not involve the position vectors of the material points, and delivers the value of the compressive load at which self-contact initiates. Furthermore, we show that one of the two integrals, namely the \emph{Hamiltonian}, persists after contact. We compute post-contact configurations of modes three through ten for a pinned-pinned buckled elastica. At a given value of the compressive load, we report multiple post-contact configurations for modes eight and nine. Finally, we show that an infinite force is required to transition from a point contact to a line contact in symmetric post-contact configurations of odd modes.

Twisted bilayer graphene near integer fillings hosts correlated single-particle excitations whose dispersion and linewidth are increasingly accessible experimentally. We study these excitations using the topological heavy-fermion model, which captures both strong correlations and band topology of twisted bilayer graphene. In the decoupled limit, where both the single-particle fc hybridization and the Hund coupling between f and c electrons are absent, the model admits exact solutions in which free Dirac fermions coexist with interacting f electrons that form zero-width Hubbard bands. By treating the fc hybridization and Hund coupling perturbatively around this solvable limit, we obtain analytical results for the single-particle self-energy. From the resulting self-energy, we derive explicit expressions for both dispersion renormalization and scattering rates of both Hubbard-band excitations and low-energy Dirac modes, thereby establishing an analytical framework for understanding correlated excitations in twisted bilayer graphene. We analyze the scattering of the two kinds, Gamma3 and Gamma1,2, of Dirac electrons and find that they arise from different mechanisms. We also briefly investigate the effect of strain. Finally, we compare these analytical expressions with DMFT results for the same model.

We investigate fermions on Carrollian manifolds. We complement previous intrinsic analysis by deriving Carrollian fermion actions from a relativistic Dirac theory via a systematic expansion in the speed of light ($c$). We then study relativistic fermions in light-cone coordinates and their connection to Carrollian fermions in one lower dimension. This follows from the recent observation that the Poincaré algebra, written in lightcone coordinates contains (two) co-dimension one Carroll sub-algebras. Our results establish a clear bridge between intrinsic Carrollian constructions, small $c$-expansion and light-cone dynamics. In the process, we understand why Carrollian fermions in $D$-dimensions have features that relate them to relativistic fermions in both $D$ and $(D+1)$ dimensions.

Exploiting enhanced sensitivity of a system in the vicinity of a phase transition boundary, critical quantum metrology to date still suffers from gap-closure related bottleneck effects, namely, critical slowing down of the sensing dynamics and a drastic shrinking of the parameter sensing window. To alleviate the said bottleneck inherent to any homogeneous lattice used for sensing, here we propose to leverage the intrinsic hopping inhomogeneity arising from bosonic ladder-operator matrix elements in Fock-space lattices (FSLs). Specifically, using a two-mode Jaynes--Cummings-type model, we show that the sensing parameter can be imprinted onto a topological zero-energy mode of the FSL. The key system parameters thus become cell dependent, effectively tracing out a curve in a topological phase diagram. Cell-dependent criticality emerges when this curve crosses or approaches a topological phase boundary, without globally tuning the lattice close to criticality. An external control parameter reshapes this curve, continuously tuning the scaling of the quantum Fisher information from the standard to the Heisenberg scaling while maintaining broad sensing coverage and a reduced gap cost. Furthermore, a local photon-number measurement on a single cavity saturates the quantum Fisher information. These results identify FSLs as a scalable and practical route to criticality-based quantum metrology.

The ongoing development of hardware that is capable of reliably executing general quantum algorithms requires quantum error-correcting codes that are both practical for realisation and rapidly reduce logical error rates as they are scaled up. Here we introduce the dynamic compass code, a code that can be implemented with a modest footprint on the heavy-hex lattice while also demonstrating a threshold. The dynamic code is obtained by choosing a novel measurement schedule for the syndrome extraction circuit of the heavy-hex subsystem code. We numerically evaluate its performance and observe that different choices of schedule can provide a trade-off in protection against logical errors in the $X$ vs $Z$ basis. We also demonstrate that this new measurement schedule provides the code with a threshold for stability experiments. We finally show how the dynamic compass code could be used for fault-tolerant logic by illustrating lattice surgery between code patches.

Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time $T$, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, $R$: when the stochastic signal exhibits high-rank correlations across the $R$ channels and the probe is entangled, the average variance (per parameter) scales no better than $Ω(1/(TR^2))$. For collective $k$-body dissipation, $R=Θ(N^k)$, signifying super-Heisenberg scaling with the system size $N$. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.

In many strongly correlated electron systems, non-Fermi liquid behavior and unconventional superconductivity can be viewed as emerging from an effective 4-fermion interaction with a singular frequency dependence. A pairing instability in such a system is qualitatively different from that in a Fermi liquid and generally gives rise to multiple pairing states with topologically distinct gap functions. However, in the systems studied so far, a topologically trivial solution has the lowest energy. Here we show that a repulsive Hubbard-type interaction with a finite cutoff added to a model with a singular dynamical interaction selects, in some parameter range, the theretofore subleading, topologically nontrivial solution. We consider a minimal model that displays this behavior and show that the transformation between the topologically trivial and nontrivial gap functions necessarily occurs via an intermediate phase with topology-induced breaking of time-reversal symmetry.

Boundary time crystals (BTCs) are prominent examples of continuous time crystals in collective spin systems governed by Lindbladian evolution. To date, their analysis has mostly relied on semiclassical and numerical approaches. Here, we develop a fully quantum-compatible framework to classify collective spin dynamics and show that BTC behavior emerges from the absence of non-trivial weak symmetries of the Liouvillian. To this end, we introduce an irreducible tensor representation of operator space, in which the Lindbladian dynamics maps onto a non-Hermitian hopping problem. Within this picture, the dynamics corresponds to the transport of operator weight across tensor sectors. This mapping allows an identification of distinct dynamical regimes, including collective precession, pure relaxation, and the BTC phase, within a single unified framework. We show that BTCs arise from non-reciprocal transport in operator space, which delocalizes Liouvillian eigenmodes across multiple tensor sectors. This non-reciprocal transport provides a microscopic mechanism for the insensitivity to initial conditions of BTC oscillations. More broadly, our results establish operator space transport as a perspective for understanding dissipative many-body dynamics and highlights connections to non-Hermitian phenomena.

Gravitational waves provide a powerful probe of both the astrophysical processes driving black hole mergers and the dynamics of the Universe, but these measurements rely on accurately inferring the unknown underlying population. We perform an agnostic reconstruction of the primary mass distribution using B-splines, characterising the emergence of structure with increasing model complexity. Using the latest gravitational-wave transient catalog, GWTC-4.0, we identify multiple mass features and find evidence suggesting a logarithmic hierarchy in the population. We show that this structure directly impacts measurements of the Hubble constant, primarily through features at the population boundaries. Finally, we introduce an approach that isolates a subpopulation of low-mass events to mitigate modelling systematics, providing a promising path toward robust population-based cosmology with future datasets.

Unlike conventional charge-2e superconductors, a charge-4e superconductor exhibits long-range coherence of electron quartets rather than Cooper pairs. Clear zero-temperature realizations of charge-4e superconductivity remain rare. Here, we investigate the zero-temperature phase diagram of the attractive SU(4) Hubbard model with numerically exact, large-scale quantum Monte Carlo (QMC) simulations overcoming major technical hurdles. We identify both charge-2e and charge-4e superconducting phases. Upon increasing interaction, charge-2e correlations are suppressed and eventually vanish, while the charge-4e correlations remain robust and converge with system size, signaling the onset of a quartet-condensed phase. Interestingly, across the charge-2e--charge-4e transition, single electrons remain gapped, while charge-2e correlations exhibit a scaling behavior inconsistent with a conventional Landau description. These features are naturally captured by a fractionalized framework in which the physical charge-2e order parameter is a composite field coupled to an emergent non-Abelian gauge structure. We formulate an Sp(4) gauge-Higgs theory that realizes deconfined quantum pseudocriticality between the Higgs (charge-2e) phase and the confined (charge-4e) phase. The Sp(4) gauge-Higgs theory yields pseudocriticality through a fixed-point collision, and its one-loop collision-point exponents quantitatively track the QMC results. Our results establish charge-4e superconductivity as a bona fide zero-temperature phase, provide a simple model for future studies in a numerically exact framework, and reveal an unconventional route to superconducting criticality.

We show that the spin conductivity of a metal approaching a ferromagnetic quantum critical point exhibits divergent fluctuation corrections. This effect arises from critical spin fluctuations and constitutes a spin analog of the Aslamazov-Larkin theory of paraconductivity in superconductors. The spin current is derived in linear response within a Gaussian-level treatment of the effective action for a system with easy-plane magnetic anisotropy. We demonstrate the consistency of our spin transport theory by showing that it (i) fulfills the Ward identity and (ii) yields vanishing spin stiffness in the normal state. The critical enhancement of the spin conductivity is interpreted as incipient spin superfluidity in the quantum critical region. This is further supported by an intuitive picture based on the current-loop representation of the easy-plane ferromagnet.

Higgs final states are prime targets in the search for physics beyond the Standard Model. In the conventional picture, $SU(2)$ symmetry together with the Goldstone Equivalence Theorem correlates Higgs and gauge-boson final states, implying comparable sensitivity in channels such as $hh$, $ZZ$, and $WW$ in searches for heavy resonances. In this work, we identify a mechanism to parametrically violate this expectation. We show that higher-order Higgs couplings can induce an electroweak-symmetry-breaking enhancement that selectively amplifies Higgs-rich final states, allowing them to become the leading discovery channels of new resonances. For scalar resonances, this can make di-Higgs the dominant bosonic signal. For resonance masses higher than a couple of TeV, it also opens resonant tri-Higgs and four-Higgs channels as well-motivated search targets. The same underlying mechanism extends to heavy fermionic and vector resonances, where it can similarly enhance channels such as $ht$, $Zh$, and $γh$. We present this framework in effective field theory, demonstrate possible UV completions, and discuss its implications for collider searches.

In the framework of Soft de Sitter Effective Theory (SdSET), the Fokker-Planck equation for the late-time dynamics of the massless minimally coupled scalar field and its extension to the Kramers-Moyal equation are obtained from operator mixing of composite operators of the effective superhorizon field. We construct the formalism for composite-operator renormalisation, mixing and matching in dimensional regularisation, allowing for computations beyond the leading order. The general formalism is illustrated in free SdSET, which already features non-trivial structures including the well-known diffusion coefficient for stochastic inflation. As explicit examples in the interacting theory, we renormalise the one-loop bispectrum and the two-loop one-point function of the composite operator $φ_+^2$, and match them onto their full-theory counterparts. These results allow us to determine the next-to-leading order (two-loop) correction to the diffusion term of the Fokker-Planck equation of stochastic inflation for the first time.

We explore the possibility that neutrinos couple to an interacting sterile sector, providing a novel portal that generalizes the heavy neutral lepton portal to a composite setting. For a low confinement scale, high-energy neutrino beams can disintegrate into collimated sprays of hidden states, referred to as dark jets. This dynamics gives rise to two characteristic signatures in high energy neutrino beams. First, long-lived dark resonances can enhance the neutral-current to charged-current ratio. Second, shorter-lived dark states produced in neutrino neutral currents can produce single or multiple displaced vertices and even emerging jets, depending on the kinematics. These signals probe regions of parameter space beyond existing constraints from meson, electroweak, and Higgs decays, as well as from searches for displaced decays at beam dump experiments. We study these phenomena within broad classes of ultraviolet completions and identify scenarios in which high-energy neutrino beams provide leading sensitivity to neutrino compositeness. Such scenarios generically induce higher-dimensional contact interactions, which we classify and study alongside their complementary experimental signatures. Finally, we outline an experimental program spanning both the intensity and energy frontiers. Near-term neutrino facilities (DUNE, FPF) and running flavor experiments (LHCb, Belle II) can probe neutrino compositeness through neutrino disintegration into dark jets and displaced B-meson decays. Future colliders, particularly the Future Circular Collider (FCC-ee), will ultimately provide the strongest sensitivity to the compositeness scale via displaced Z decays.

Tidal stellar shells and streams are two of the most intriguing low-surface-brightness features within galaxies, consisting of stars accreted from satellite galaxies. A crucial ingredient in determining which type of feature will be formed is the orbit of the satellite galaxy. Additionally, the distribution of stars from these satellite galaxies within the merger remnant and the original location of these stars within the progenitor satellite galaxy provide important clues about the deposition of the stellar component in the resulting galaxy. We utilize the cosmological hydrodynamical simulation Magneticum Pathfinder and expand on the work by Valenzuela & Remus (2024) and Stoiber et al. (2025) to present a case study for the formation of a stream and a shell system. We analyze their orbits and the distributions of stellar particles within their host galaxy and compare them to their initial location within the progenitor satellite galaxy. We find that the orbit of the stream progenitor is more circular than the progenitor of the shell system. The stellar particles of the stream from different initial radii are found at roughly the same distances with respect to the host galaxy. However, the part of the stream visible in mock observations - not hidden by the host galaxy - consists of stars from within the core of the progenitor ($r/r_{1/2} < 1$). On the other hand, the stellar particles of the shell system retain their radial ordering: Stars that were initially at small radii in the satellite galaxy also remain closer to the center of the host galaxy.

The path-integral approach to the double well has long been limited by the dilute instanton gas approximation. We show that if the finite Euclidean-time structure is taken seriously by using exact saddles, the dilute gas can be sidestepped, allowing the partition function and energy levels to be computed systematically. At each instanton order, the full resurgent structure -- which saddles contribute, what asymptotic growth is expected and how ambiguities cancel -- is encoded in a finite-dimensional Picard--Lefschetz contour integral over the quasi-zero modes with a clear geometric interpretation. Working at finite $T$ is essential: the dilute instanton gas can only access the ground-state splitting, whereas the exact finite-$T$ computation systematically produces the non-perturbative energy splittings for all excited states, including their full dependence on the level number. The key ingredients -- Weierstrass elliptic functions for the saddles, Lamé operators for the fluctuations and Picard--Fuchs equations for the periods -- form a coherent mathematical framework that both overlaps and complements that of Exact WKB.

We develop a controlled theoretical framework for the topological heavy fermion model relevant to magic-angle twisted bilayer graphene, where low density conduction electrons hybridize with a lattice of strongly interacting f-sites. By tracing out the localized electrons, we derive an effective action for the conduction electrons with long-range in time effective interactions, built from correlators of the single f-site problem. We identify a small hybridization-phase-space parameter resulting in a controlled loop expansion, enabling the derivation of nonperturbative results in either the interaction or the hybridization strength. To tree-level, the results are equivalent to the Hubbard I approximation. At higher loop order, we derive two key results applicable to temperatures above the flavor ordering temperature and below the on-site charging energy: 1) the quasi-particle lifetime, 2) the flavor susceptibility of the system. Remarkably, despite being strongly interacting, we find the susceptibility to accurately obey a Curie-Weiss law parametrically close to the Curie temperature.

We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all $n$ modes squeezed. For random brickwall circuits, we show that entanglement, as measured by the Rényi-2 entropy, grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in $L^2$ Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability. The corresponding Gaussian unitary $\tilde{\mathcal U}$ for the approximate implementation retains high output fidelity $|\langleψ|\mathcal U^\dagger \tilde{\mathcal U}|ψ\rangle|^2$ for pure states $|ψ\rangle$ with constrained expected photon-number.

HWO aims to directly image objects orbiting Sun-like stars, using a 6-m telescope capable of high-contrast imaging ($10^{-10}$) and spectroscopy to search for biosignatures in planets located in the habitable zone. Recent laboratory demonstrations and ground-based telescope projects have shown the effectiveness of SMFs in spectroscopy, paving the way for SMF-fed spectrographs in future space missions like HWO. SMFs enhance spectral stability and reduce modal noise. HWO spectroscopy will need extended integration times, potentially lasting weeks. During these observations, the wavefront must be precisely measured and maintained to achieve the deep contrast and robust calibration of starlight contamination necessary for exoplanet characterization. We show that photonic lanterns (PLs) are ideally suited to meet these requirements. PLs are compact devices that couple light over a broader angular range than SMFs, ensuring higher throughput, converting a multimode input into multiple single-mode outputs. Positioned at the focal plane, they measure the complex amplitude of the coherent starlight within $\sim$ 2 l/D of the planet image, acting as compact wavefront sensors. Among the different variants of PLs that have emerged, the Hybrid-Mode Selective Photonic Lantern (HMSPL) is particularly attractive, as it directs object light into a central SMF feeding a mid-R spectrograph for exoplanet spectroscopy, while the adjacent SMFs route surrounding speckle light to a low-R spectrograph for rapid wavefront sensing. This dual function eliminates non-common path aberrations, optimizing injection efficiency and background suppression. We introduce HMSPL's dual role and planned tests at UTSA's high-contrast imaging lab and at SCExAO at the Subaru Telescope.