Finite groups have more conjugacy classes

We prove that for every $\epsilon > 0$ there exists a $\delta > 0$ so that every group of order $n \geq 3$ has at least $\delta \log_{2} n/{(\log_{2} \log_{2} n)}^{3+\epsilon}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order $n$ has more than $\log_{3}n$ conjugacy classes. We answer Bertram's question in the affirmative for groups with a trivial solvable radical

On a conjecture of Gluck

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.Comment: 16 page

A phason disordered two dimensional quantum antiferromagnet

We examine a novel type of disorder in quantum antiferromagnets. Our model consists of localized spins with antiferromagnetic exchanges on a bipartite quasiperiodic structure, which is geometrically disordered in such a way that no frustration is introduced. In the limit of zero disorder, the structure is the perfect Penrose rhombus tiling. This tiling is progressively disordered by augmenting the number of random "phason flips" or local tile-reshuffling operations. The ground state remains N\'eel ordered, and we have studied its properties as a function of increasing disorder using linear spin wave theory and quantum Monte Carlo. We find that the ground state energy decreases, indicating enhanced quantum fluctuations with increasing disorder. The magnon spectrum is progressively smoothed, and the effective spin wave velocity of low energy magnons increases with disorder. For large disorder, the ground state energy as well as the average staggered magnetization tend towards limiting values characteristic of this type of randomized tilings.Comment: 5 pages, 7 figure

Geometry fluctuations in a two-dimensional quantum antiferromagnet

The paper considers the effects of random fluctuations of the local spin connectivities (fluctuations of the geometry) on ground state properties of a two-dimensional quantum antiferromagnet. We analyse the behavior of spins described by the Heisenberg model as a function of what we call phason flip disorder, following a terminology used for aperiodic systems. The calculations were carried out both within linear spin wave theory and using quantum Monte Carlo simulations. An "order by disorder" phenomenon is observed in this model, wherein antiferromagnetism is found to be enhanced by phason disorder. The value of the staggered order parameter increases with the number of defects, accompanied by an increase in the ground state energy of the system.Comment: 5 pages, 7 figures. Shortened and corrected version (as accepted for publication in Physical Review B

Vertex dynamics during domain growth in three-state models

Topological aspects of interfaces are studied by comparing quantitatively the evolving three-color patterns in three different models, such as the three-state voter, Potts and extended voter models. The statistical analysis of some geometrical features allows to explore the role of different elementary processes during distinct coarsening phenomena in the above models.Comment: 4 pages, 5 figures, to be published in PR

Revealing signatures of planets migrating in protoplanetary discs with ALMA multi-wavelength observations

Recent observations show that rings and gaps are ubiquitous in protoplanetary discs. These features are often interpreted as being due to the presence of planets; however, the effect of planetary migration on the observed morphology has not been investigated hitherto. In this work we investigate whether multiwavelength mm/submm observations can detect signatures of planet migration, using 2D dusty hydrodynamic simulations to model the structures generated by migrating planets and synthesising ALMA continuum observations at 0.85 and 3 mm. We identify three possible morphologies for a migrating planet: a slowly migrating planet is associated with a single ring outside the planet's orbit, a rapidly migrating planet is associated with a single ring inside the planet's orbit while a planet migrating at intermediate speed generates one ring on each side of the planet's orbit. We argue that multiwavelength data can distinguish multiple rings produced by a migrating planet from other scenarios for creating multiple rings, such as multiple planets or discs with low viscosity. The signature of migration is that the outer ring has a lower spectral index, due to larger dust grains being trapped there. Of the recent ALMA observations revealing protoplanetary discs with multiple rings and gaps, we suggest that Elias 24 is the best candidate for a planet migrating in the intermediate speed regime.Comment: Accepted for publication in MNRA

Epidemic spreading in evolving networks

A model for epidemic spreading on rewiring networks is introduced and analyzed for the case of scale free steady state networks. It is found that contrary to what one would have naively expected, the rewiring process typically tends to suppress epidemic spreading. In particular it is found that as in static networks, rewiring networks with degree distribution exponent $\gamma >3$ exhibit a threshold in the infection rate below which epidemics die out in the steady state. However the threshold is higher in the rewiring case. For $2<\gamma \leq 3$ no such threshold exists, but for small infection rate the steady state density of infected nodes (prevalence) is smaller for rewiring networks.Comment: 7 pages, 7 figure