#### Journal of Physics A: Mathematical and Theoretical

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ISSN / EISSN
:
1751-8113 / 1751-8121

Published by: IOP Publishing
(10.1088)

Total articles ≅ 11,794

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Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac6070

**Abstract:**

For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path (NP) operator and thus a continuous family of one-point functions ${W}_{k}\equiv \u27e8\mathcal{R}\cdot {k}^{\ell}\u27e9$ , where ℓ is the number of independent (i.e., non-overlapping) nested closed paths surrounding the center, k is a path fugacity, and $\mathcal{R}$ projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling ${W}_{k}\sim {L}^{-{X}_{\text{NP}}}$ , with L the linear system size, and we determine the exponent X ${X}_{\text{NP}}(k)=\frac{3}{4}{\varphi}^{2}-\frac{5}{48}{\varphi}^{2}/({\varphi}^{2}-\frac{2}{3})$ with k = 2 cos(πϕ), which reproduces the exact results for k = 0, 1 and agrees with the high-precision estimate of ${X}_{\text{NP}}$ for other k values. In addition, we observe that W

_{NP}as a function of k. On the basis of our numerical results, we conjecture an analytical formula,_{2}(L) = 1 for site percolation on the triangular lattice with any size L, and we prove this identity for all self-matching lattices.
Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac6240

**Abstract:**

This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether’s theorem and other theorem characterizing the Lie subalgebra of symmetries of a forced discrete Lagrangian system. Moreover, we develop a Hamilton–Jacobi theory for forced discrete Hamiltonian systems. These results are useful for the construction of so-called variational integrators, which, as we illustrate with some examples, are remarkably superior to the usual numerical integrators such as the Runge–Kutta method.

Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac62ba

**Abstract:**

In this paper we provide a formula for the canonical differential form of the hypersimplex Δ ${\mathcal{M}}_{n,k}$ to m = 2, which has been conjectured to share many properties with the hypersimplex, and we provide counterexamples for these conjectures. Nevertheless, we find interesting momentum amplituhedron-like logarithmic differential forms in the m = 2 version of the spinor helicity space, that have the same singularity structure as the hypersimplex canonical forms.

_{k,n}for all n and k. We also study the generalization of the momentum amplituhedron
Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac5f7a

**Abstract:**

A one-dimensional, driven lattice gas with a freely moving, driven defect particle is studied. Although the dynamics of the defect are simply biased diffusion, it disrupts the local density of the gas, creating nontrivial nonequilibrium steady states. The phase diagram is derived using mean field theory and comprises three phases. In two phases, the defect causes small localized perturbations in the density profile. In the third, it creates a shock, with two regions at different bulk densities. When the hopping rates satisfy a particular condition (that the products of the rates of the gas and defect are equal), it is found that the steady state can be solved exactly using a two-dimensional matrix product ansatz. This is used to derive the phase diagram for that case exactly and obtain exact asymptotic and finite size expressions for the density profiles and currents in all phases. In particular, the front width in the shock phase on a system of size L is found to scale as L

^{1/2}, which is not predicted by mean field theory. The results are found to agree well with Monte Carlo simulations.
Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac5a90

**Abstract:**

Anomalous diffusion with a power-law time dependence $\u27e8|\mathbf{R}{|}^{2}(t)\u27e9\simeq {t}^{{\alpha}_{i}}$ of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents α

_{i}. Here we consider the case when such a crossover occurs from a first regime with α_{1}to a second regime with α_{2}such that α_{2}> α_{1}, i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann–Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green’s function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Lévy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes.
Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac5e75

**Abstract:**

Many processes in cell biology involve diffusion in a domain Ω that contains a target $\mathcal{U}$ whose boundary $\partial \mathcal{U}$ is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. In this paper we extend the theory of diffusion-mediated absorption to cases where the whole interior target domain $\mathcal{U}$ acts as a partial absorber rather than the target boundary $\partial \mathcal{U}$ . Now the particle can freely enter and exit $\mathcal{U}$ , and is only able to react (be absorbed) within $\mathcal{U}$ . The appropriate Brownian functional is then the occupation time (accumulated time that the particle spends within $\mathcal{U}$ ) rather than the boundary local time. We show that both cases can be considered within a unified framework, which consists of a boundary value problem (BVP) for the propagator of the corresponding Brownian functional and an associated stopping condition. We illustrate the theory by calculating the mean first passage time (MFPT) for a spherical target $\mathcal{U}$ located at the center of a spherical domain Ω. This is achieved by solving the propagator BVP directly, rather than using spectral methods. We find that if the first moment of the stopping time density is infinite, then the MFPT is also infinite, that is, the spherical target is not sufficiently absorbing.

Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac5fe8

**Abstract:**

We study the time evolution of the spin-1/2 XXZ chain initialized in a domain wall state, where all spins to the left of the origin are up, all spins to its right are down. The focus is on exact formulae, which hold for arbitrary finite (real or imaginary) time. In particular, we compute the amplitudes corresponding to the process where all but k spins come back to their initial orientation, as a k-fold contour integral. These results are obtained using a correspondence with the six vertex model, and taking a somewhat complicated Hamiltonian/Trotter-type limit. Several simple applications are studied and also discussed in a broader context.

Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac61b7

**Abstract:**

I describe the interplay between Minkowski and Euclidean signature gamma matrices, Majorana fermions, and discrete and continuous symmetries in all spacetime dimensions. In particular I argue that the space-time dimensions in which various classes of Majorana fermions exists are the same in both Minkowski and Euclidean signature.

Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac62b9

**Abstract:**

The semiclassical structure of resonance states of classically chaotic scattering systems with partial escape is investigated. We introduce a local randomization on phase space for the baker map with escape, which separates the smallest multifractal scale from the scale of the Planck cell. This allows for deriving a semiclassical description of resonance states based on a local random vector model and conditional invariance. We numerically demonstrate that the resulting classical measures perfectly describe resonance states of all decay rates γ for the randomized baker map. By decreasing the scale of randomization these results are compared to the deterministic baker map with partial escape. This gives the best available description of its resonance states. Quantitative differences indicate that a semiclassical description for deterministic chaotic systems must take into account that the multifractal structures persist down to the Planck scale.

Journal of Physics A: Mathematical and Theoretical, Volume 55; https://doi.org/10.1088/1751-8121/ac5e74

**Abstract:**

We extend the matrix-resolvent method for computing logarithmic derivatives of tau-functions to the Ablowitz–Ladik hierarchy. In particular, we derive a formula for the generating series of the logarithmic derivatives of an arbitrary tau-function in terms of matrix resolvents. As an application, we provide a way of computing certain integrals over the unitary group.