Shift in critical temperature for random spatial permutations with cycle weights

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter $\alpha$. For weak interactions, the shift in critical temperature is expected to be linear in $\alpha$ with constant of linearity $c$. Using Markov chain Monte Carlo methods and finite-size scaling, we find $c = 0.618 \pm 0.086$. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations.Comment: v2 incorporated reviewer comments. v3 removed two extraneous figures which appeared at the end of the PDF

Ground state at high density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and Likos et al. (2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superposed on the sites of a close-packed lattice.Comment: Note adde

Crystalline ground states for classical particles

Pair interactions whose Fourier transform is nonnegative and vanishes above a wave number K_0 are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in any dimension d. A typical three dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The result is obtained for densities rho >= rho_d where rho_1=K_0/2pi, rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is a unique periodic GSC which is the uniform chain, the triangular lattice and the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density rho with all reciprocal lattice vectors not smaller than K_0, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6 sqrt{3})(K_0/pi)^3.Comment: final versio

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\"odinger dynamics generated by the Fibonacci Hamiltonian.Comment: 23 page

H\"older Continuity of the Integrated Density of States for the Fibonacci Hamiltonian

We prove H\"older continuity of the integrated density of states for the Fibonacci Hamiltonian for any positive coupling, and obtain the asymptotics of the H\"older exponents for large and small couplings.Comment: 18 page

Quality and use of habitat patches by wild boar (Sus scrofa) along an urban gradient

Expansion and urbanization process of wild boar (Sus scrofa) populations lead to serious human–wildlife conflicts in many cities, e.g. in Budapest, Hungary. In this study we evaluated the penetration potential of the species into the inner urban areas by identifying the occurrence of wild boar and the quality of the habitat patches for them along an urban gradient from the periphery towards the centre. Wild boar rooting intensity, shrub cover and the availability of woody species giving favourable food to wild boar were measured in four different habitat patches. The availability of hiding shrub patches was much higher in the outer areas than in the inner ones. Similarly, the proportion of shrub and tree species providing favourable food for wild boar decreased towards the centre. Accordingly, we found rooting only in two areas nearer to the city boundary. Based on our results at the peripheral areas permanent presence of wild boar in near-natural habitats should be expected, but not in the inner green zones. We recommend to monitor the urban wild boar presence and evaluate the quality of urban green patches to mitigate problems related to the wild boars

A mechanical model of normal and anomalous diffusion

The overdamped dynamics of a charged particle driven by an uniform electric field through a random sequence of scatterers in one dimension is investigated. Analytic expressions of the mean velocity and of the velocity power spectrum are presented. These show that above a threshold value of the field normal diffusion is superimposed to ballistic motion. The diffusion constant can be given explicitly. At the threshold field the transition between conduction and localization is accompanied by an anomalous diffusion. Our results exemplify that, even in the absence of time-dependent stochastic forces, a purely mechanical model equipped with a quenched disorder can exhibit normal as well as anomalous diffusion, the latter emerging as a critical property.Comment: 16 pages, no figure

Spectra of Discrete Schr\"odinger Operators with Primitive Invertible Substitution Potentials

We study the spectral properties of discrete Schr\"odinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to $1$ as coupling constant $\lambda$ tends to $0$. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to $\lambda$. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to $1$ as $\lambda$ tends to $0$

Tight Binding Hamiltonians and Quantum Turing Machines

This paper extends work done to date on quantum computation by associating potentials with different types of computation steps. Quantum Turing machine Hamiltonians, generalized to include potentials, correspond to sums over tight binding Hamiltonians each with a different potential distribution. Which distribution applies is determined by the initial state. An example, which enumerates the integers in succession as binary strings, is analyzed. It is seen that for some initial states the potential distributions have quasicrystalline properties and are similar to a substitution sequence.Comment: 4 pages Latex, 2 postscript figures, submitted to Phys Rev Letter

Commensurate and incommensurate correlations in Haldane gap antiferromagnets

We analyze the onset of incommensurabilities around the VBS point of the S=1 bilinear-biquadratic model. We propose a simple effective field theory which is capable of reproducing all known properties of the commensurate-incommensurate transition at the disorder point $\theta_{\rm vbs}$. Moreover, the theory predicts another special point $\theta_{\rm disp}$, distinct from the VBS point, where the Haldane gap behaves singularly. The ground state energy density is an analytic function of the model parameters everywhere, thus we do not have phase transitions in the conventional sense.Comment: 8 pages, 2 figures, to appear in PR