We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of non-intersecting paths. The starting point of our analysis is a formula for the correlation kernel due to Duits and Kuijlaars which involves the Christoffel-Darboux kernel of a particular family of non-Hermitian orthogonal polynomials. Our main results are split into two parts: the first part concerns the family of orthogonal polynomials, and the second concerns the behavior of the boundary of the so-called arctic curve. In the first half, we identify the orthogonal polynomials as a non-standard instance of little $q$-Jacobi polynomials and compute their large degree asymptotics in the $q \to 1$ regime. A consequence of this analysis is a proof that the zeros of the orthogonal polynomials accumulate on an arc of a circle and an asymptotic formula for the Christoffel-Darboux kernel. In the second half, we use these asymptotics to show that the boundary of the liquid region converges to the Airy process, in the sense of finite dimensional distributions, away from the boundary of the hexagon. At inflection points of the arctic curve, we show that we do not need to subtract/add a parabola to the Airy line ensemble, and this effect persists at distances which are $o(N^{-2/9})$ in the tangent direction.

In a previous work of the first authors, a non-holonomic model, generalising the micromorphic models and allowing for curvature (disclinations) to arise from the kinematic values, was presented. In the present paper, a generalisation of the classical models of Euler-Bernoulli and Timoshenko bending beams based on the mentioned work is proposed. The former is still composed of only one unidimensional scalar field, while the later introduces a third unidimensional scalar field, correcting the second order terms. The generalised Euler-Bernoulli beam is then shown to exhibit curvature (i.e. disclinations) linked to a third order derivative of the displacement, but no torsion (dislocations). Parallelly, the generalised Timoshenko beam is shown to exhibit both curvature and torsion, where the former is linked to the non-holonomy introduced in the generalisation. Lastly, using variational calculus, asymptotic values for the value taken by the curvature in static equilibrium are obtained when the second order contribution becomes negligible; along with an equation for the torsion in the generalised Timoshenko beam.

The propagation of nonlinear and dispersive waves in various materials can be described by the well-known Kadomtsev-Petviashvili (KP) equation, which is a (2+1)-dimensional partial differential equation. In this paper, we show that the KP equation can be used to describe the in-plane motion of compressible elastic solids with dispersion. Furthermore, a modified KP equation with cubic nonlinearity is obtained in the case of incompressible solids with dispersion. Then, several solutions of these partial differential equations are discussed and computed using a Fourier spectral method. In particular, both equations admit solitary wave solutions.

Based on a baryogenesis mechanism originating from the theory of causal fermion systems, we analyze its main geometric and analytic features in conformally flat spacetimes. An explicit formula is derived for the rate of baryogenesis in these spacetimes, which depends on the mass $m$ of the particles, the conformal factor $\Omega$ and a future directed timelike vector field $u$ (dubbed the regularizing vector field). Our analysis covers Friedmann-Lema{\^i}tre-Robertson-Walker, Milne and Milne-like spacetimes. It sets the ground for concrete, quantitative predictions for specific cosmological spacetimes.

In this work we consider the $N$-body Hamiltonian describing the microscopic structure of a quantum gas of almost-bosonic anyons. This description includes both extended magnetic flux and spin-orbit/soft-disk interaction between the particles which are confined in a scalar trapping potential. We study a physically well-motivated ansatz for a sequence of trial states, consisting of Jastrow repulsive short-range correlations and a condensate, with sufficient variational freedom to approximate the ground state (and possibly also low-energy excited states) of the gas. In the limit $N \to \infty$, while taking the relative size of the anyons to zero and the total magnetic flux $2\pi\beta$ to remain finite, we rigorously derive the stationary Chern-Simons-Schr\"odinger/average-field-Pauli effective energy density functional for the condensate wave function. This includes a scalar self-interaction parameter $\gamma$ which depends both on $\beta$, the diluteness of the gas, and the spin-orbit coupling strength $g$, but becomes independent of these microscopic details for a particular value of the coupling $g=2$ in which supersymmetry is exhibited (on all scales, both microscopic and mesoscopic) with $\gamma=2\pi|\beta|$. Our findings confirm and clarify the predictions we have found in the physics literature.

Since its introduction, the Potts model has gained widespread popularity across various fields due to its diverse applications. Even minor advancements in this model continue to captivate scientists worldwide, and small modifications often intrigue researchers from different disciplines. This paper investigates a one-dimensional \(q\)-state modified Potts model influenced by an external magnetic field. By leveraging the transfer matrix method, exact expressions are derived for key thermodynamic quantities, including free energy, entropy, magnetization, susceptibility, and specific heat capacity. Numerical analyses explore how these thermodynamic functions vary with relevant parameters, offering insights into the system's behavior. Additionally, the asymptotic properties of these quantities are examined in the limiting cases \(T \to 0\) and \(T \to \infty\). The findings contribute to a deeper understanding of the model's thermodynamic characteristics and highlight its potential applications across various disciplines.

It is shown that the existence of a local conserved charge supported by three neighboring sites, or its local version, Reshetikhin's condition, suffices to guarantee the existence of all higher conserved charges and hence the integrability of a quantum spin chain. This explains the ``coincidence'' that no counterexample is known to Grabowski and Mathieu's long-standing conjecture despite the folklore that the conservation of local charges of order higher than 4 imposes additional constraints not implied by the conservation of the three-local charge.

A convergent power series solution is obtained for the SIR model, using an asymptotically motivated gauge function. For certain choices of model parameter values, the series converges over the full physical domain (i.e., for all positive time). Furthermore, the radius of convergence as a function of nondimensionalized initial susceptible and infected populations is obtained via a numerical root test.

We study local, higher-spin conserved currents in integrable $2d$ sigma models that have been deformed via coupling to auxiliary fields. These currents generate integrability-preserving flows introduced by Smirnov and Zamolodchikov. For auxiliary field (AF) deformations of a free boson, we prove that local spin-$n$ currents exist for all $n$ and give recursion relations that characterize Smirnov-Zamolodchikov (SZ) flows driven by these currents. We then show how to construct spin-$2n$ currents in a unified class of auxiliary field sigma models with common structure -- including AF theories based on the principal chiral model (PCM), its non-Abelian T-dual, (bi-)Yang-Baxter deformations of the PCM, and symmetric space models -- for interaction functions of one variable, and describe SZ flows driven by any function of the stress tensor in these cases. Finally, we give perturbative solutions for spin-$3$ SZ flows in any member of our unified class of AF models with underlying $\mathfrak{su}(3)$ algebra. Part of our analysis shows that the class of AF deformations can be extended by allowing the interaction function to depend on a larger set of variables than has previously been considered.

In this paper, we establish a relation between the quantum corner VOA $q\widetilde{Y}_{L,0,N}[\Psi]$, which can be regarded as a generalization of quantum $W_N$ algebra, and Sergeev-Veselov super Macdonald polynomials. We demonstrate precisely that, under a specific map, the correlation functions of the currents of $q\widetilde{Y}_{L,0,N}[\Psi]$, coincide with the Sergeev-Veselov super Macdonald polynomials.

We revisit the Markov Entropy Decomposition, a classical convex relaxation algorithm introduced by Poulin and Hastings to approximate the free energy in quantum spin lattices. We identify a sufficient condition for its convergence, namely the decay of the effective interaction. We prove that this condition is satisfied for systems in 1D at any temperature as well as in the high-temperature regime under a certain commutativity condition on the Hamiltonian. This yields polynomial and quasi-polynomial time approximation algorithms in these settings, respectively. Furthermore, the decay of the effective interaction implies the decay of the conditional mutual information for the Gibbs state of the system. We then use this fact to devise a rounding scheme that maps the solution of the convex relaxation to a global state and show that the scheme can be efficiently implemented on a quantum computer, thus proving efficiency of quantum Gibbs sampling under our assumption of decay of the effective interaction.

These lectures present a brief introduction to measurement theory for QFT in possibly curved spacetimes introduced by the author and R. Verch [Comm. Math. Phys. 378 (2020) 851-889]. Topics include: a brief introduction to algebraic QFT, measurement schemes in QFT, state updates, multiple measurements and the resolution of Sorkin's "impossible measurement" problem. Examples using suitable theories based on Green hyperbolic operators are given, and the interpretational significance of the framework is briefly considered. The basic style is to give details relating to QFT while taking for granted various facts from the theory of globally hyperbolic spacetimes.

We investigate the Cauchy problem for the spin-1 Gross-Pitaevskii(GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng $etal.$ (Commun. Math. Phys. 382, 585-611 (2021)) reported the long-time asymptotic result with error $\mathcal{O}(\frac{\log t}t)$ for the spin-1 GP equation that only exists in the continuous spectrum. The main purpose of our work is to further generalize and improve Geng's work. Compared with the previous work, our asymptotic error accuracy has been improved from $\mathcal{O}(\frac{\log t}t)$ to $\mathcal{O}(t^{-3/4})$. More importantly, by establishing two matrix valued functions, we obtained effective asymptotic errors and successfully constructed asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including two cases: (i)coexistence of discrete and continuous spectrum; (ii)only continuous spectrum which considered by Geng's work with error $\mathcal{O}(\frac{\log t}t)$. For the case (i), the corresponding asymptotic approximations can be characterized with an $N$-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$. For the case (ii), the corresponding asymptotic approximations can be characterized with the leading term on the continuous spectrum and the residual error order $\mathcal{O}(t^{-3/4})$. Finally, our results confirm the soliton resolution conjecture for the spin-1 GP equation.

We present a unified free field realization of representations for the quantum toroidal algebra of $\mathfrak{gl}_1$ with arbitrary levels, constructed using six free boson fields. This realization arises from a specialized factorization of the structure function within the defining relations of the quantum toroidal algebra of $\mathfrak{gl}_1$. Utilizing this free field realization, we further develop intertwining operators for the algebra of $\mathfrak{gl}_1$.

The orbifold/condensation completion procedure of defect topological quantum field theories can be seen as carrying out a lattice or state sum model construction internal to an ambient theory. In this paper, we propose a conceptual algebraic description of orbifolds/condensations for arbitrary tangential structures in terms of higher dagger structures and higher idempotents. In particular, we obtain (oriented) orbifold completion from (framed) condensation completion by using a general strictification procedure for higher dagger structures which we describe explicitly in low dimensions; we also discuss the spin and unoriented case. We provide several examples of higher dagger categories, such as those associated to state sum models, (orbifolds of) Landau--Ginzburg models, and truncated affine Rozansky--Witten models. We also explain how their higher dagger structures are naturally induced from rigid symmetric monoidal structures, recontextualizing and extending results from the literature.

We present new solutions to the Klein-Gordon equation for a scalar particle in a black string spacetime immersed in an anisotropic quintessence fluid surrounded by a cloud of strings, extending the analysis presented in our previous work. These novel solutions are dependent on the quintessence state parameter, $\alpha_{Q}$, and are now valid for a much larger domain of the radial coordinate. We investigate the cases when $\alpha_{Q} = 0,\,1/2,\,1$, encompassing both black hole and horizonless scenarios. We express the resulting radial wave functions using the confluent and biconfluent Heun functions, with special cases represented by Bessel functions. We derive restrictions on the allowed quantum energy levels by imposing constraints on the Heun parameters to ensure polynomial solutions. Furthermore, we investigate the emergence of "dark phases" associated with the radial wave function, focusing on the interesting case of $\alpha_{Q} = 1$. Our findings provide insights into the dynamics of scalar particles in this complex spacetime and the potential impact of dark energy on quantum systems.