Table of contents for issue 45, volume 52, Journal of Physics A: Mathematical and Theoretical

Phylogenetics is the suite of mathematical and computational methods which provide biologists with the means to infer past evolutionary relationships between observed species. The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years, as the foundations on which the practical implementations are based. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the toolkit of that tradition—techniques of groups and representations to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries; extensions to coalgebraic properties of the generators associated to rate matrices underlying the models, and possibilities to marry these with the graphical structures (trees and networks) which form the search space for inferring evolutionary trees.

Particular aspects which we wish to present to a readership accustomed to thinking from physics, include relating model classes to structural data on relevant matrix Lie algebras, as well as using manipulations with group characters (especially the operation of plethysm, for computing tensor powers) to enumerate various natural polynomial invariants, which can be enormously helpful in tying down robust, low-parameter quantities for use in inference (some of which have only come to light through our perspective). Above all, we wish to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. In some instances, well-known objects such as the Cayley hyperdeterminant (the ‘tangle’) can be directly imported into the formalism—in this case, for models with binary character traits, and for providing information about triplets of taxa. In other cases new objects appear, such as the remarkable ‘squangle’ invariants for quartet tree discrimination, which for DNA data are of quintic degree, with their own unique interpretation in the phylogenetic modelling context. All this hints strongly at the natural and universal presence of entanglement as a phenomenon which reaches across disciplines. We hope that this broad perspective may in turn furnish new insights of use in physics.

In this note we approach the classical, Newtonian, gravitational N-body problem by mean of a new, original numerical integration method. After a short summary of the fundamental characteristics of the problem, including a sketch of some of its mathematical and numerical issues, we present the new algorithm, which is applied to a set of sample cases of initial conditions in the ‘intermediate’ N regime (N  =  100). This choice of N is not due to algorithm limitation but just for computational convenience, in what this preliminary work aims mainly to the presentation of the new method and so we wanted just to provide an acceptable although statistical significant comparison with other integration schemes. The proposed algorithm seems to be fast and precise at the same time and so promising for further, more realistic, tests and scientific applications.

In this contribution we further study the classical disordered p  =  2 spherical model with Hamiltonian dynamics, or in integrable systems terms, the Neumann model, in the infinite size limit. We summarise the asymptotic results that some of us presented in a recent publication, and we deepen the analysis of the pre-asymptotic dynamics. We also discuss the possible description of the asymptotic steady state with a generalised Gibbs ensemble.

Axionic holographic RG flow solutions are studied in the context of general Einstein-axion-dilaton theories. A non-trivial axion profile is dual to the (non-perturbative) running of the -term for the corresponding instanton density operator. It is shown that a non-trivial axion solution is incompatible with a non-trivial (holographic) IR conformal fixed point. Imposing a suitable axion regularity condition allows to select the IR geometry in a unique way. The solutions are found analytically in the asymptotic UV and IR regimes, and it is shown that in those regimes the axion backreaction is always negligible. The axion backreaction may become important in the intermediate region of the bulk. To make contact with the axion probe limit solutions, a systematic expansion of the solution is developed. Several concrete examples are worked out numerically. It is shown that the regularity condition always implies a finite allowed range for the axion source parameter in the UV. This translates into the existence of a finite (but large) number of saddle-points in the large N_c limit. This ties in well with axion-swampland conjectures.

Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the ‘jump lengths’—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.

The properties of zero modes in particle-hole symmetric systems are analyzed in the presence of strong random scattering by a disordered environment. The study is based on the calculation of the time-averaged density distribution on a lattice. In particular, a flat distribution is found for strong random scattering. This result is compared with a decaying distribution for weak random scattering by an analysis of the scattering paths. In the calculation we consider the invariant measure of the average two-particle Green’s function, which is related to lattice-covering self-avoiding (LCSA) strings. In particular, strong scattering is associated with LCSA loops, whereas weaker scattering is associated with open LCSA strings. Our results are a generalization of the delocalized state observed at the band center of a one-dimensional tight-binding model with random hopping by Dyson (1953 Phys. Rev. 92 1331).

We consider overdetermined systems of difference equations for a single function u which are consistent, and propose a general framework for their analysis. The integrability of such systems is defined as the existence of higher order symmetries in both lattice directions and various examples are presented. Two hierarchies of consistent systems are constructed, the first one using lattice paths and the second one as a deformation of the former. These hierarchies are integrable and their symmetries are related via Miura transformations to the Bogoyavlensky and the discrete Sawada–Kotera lattices, respectively.

Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley’s methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the ‘probabilistic quadrant’ , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables.

It is an open question if there are leakage-free entangling Fibonacci braiding gates. In this article, we give a construction of a large family of leakage-free braiding gates which are then proved to be non-entangling. We also conducted brute-force numerical searches for braids with a word-length up to seven and found no leakage-free entangling gates. These suggest the negative for the conjecture. On the other hand, we provide a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature.

We introduce an analytical XX spin chain with asymmetrical transport properties. It has an even number N  +  1 of sites labeled by . It does not exhibit perfect state transfer (PST) from end-to-end but rather from the first site to the next to last one. In fact, PST of one-excitation states takes place between the even sites: , ; while states localized at a single odd site undergo fractional revival (FR) over odd sites only. Perfect return is witnessed at double the PST/FR time. The couplings and local magnetic fields are related to the recurrence coefficients of the dual  −1 Hahn polynomials.

In this paper, we investigate a five-dimensional Dirac fermion on a quantum graph that consists of a single vertex and N loops. We find that the model possesses a rich structure of boundary conditions for wavefunctions on the quantum graph and they can be classified into distinct categories. It is then shown that there appear degenerate four-dimensional chiral massless fermions in the four-dimensional mass spectrum. We briefly discuss how our model could naturally solve the problems of the fermion generation, the fermion mass hierarchy and the origin of the CP-violating phase.