Table of contents

We analyze the anonymous communication patterns of 2.5 million customers of a Belgian mobile phone operator. Grouping customers by billing address, we build a social network of cities that consists of communications between 571 cities in Belgium. We show that inter-city communication intensity is characterized by a gravity model: the communication intensity between two cities is proportional to the product of their sizes divided by the square of their distance.

In experiments involving dip coating flows on an infinite flat substrate which is withdrawn from an infinite liquid bath, the thin film deposited far up on the plate usually thickens in the presence of insoluble interfacial surfactant. Using perturbation analysis within the lubrication approximation we prove that the film thickens in the presence of interfacial surfactant for low capillary numbers if surface tension away from the transition and meniscus regions increases in the direction of withdrawal of the plate, a condition that should truly emerge from the solution of the full problem. Thus, we essentially show that fine scale properties of the interfacial dynamics and the dynamics in the bulk of the fluid near the transition and meniscus regions are, in fact, not important. We show that it is only the surface tension gradient far away from the transition and meniscus regions that matters. This result is arrived at by first deriving upper and lower bounds on the film thickness in terms of Marangoni and capillary numbers. An estimate based on these results and interfacial surfactant dynamics also yields a qualitative profile of the interfacial surfactant concentration that results in an increase in film thickness.

The evolution pattern of level crossings and exceptional points is studied in a non-integrable pairing model with two different integrable limits. One of the integrable limits has two independent parameter-dependent integrals of motion. We demonstrate, and illustrate in our model, that quantum integrability of a system with more than one parameter-dependent integral of motion is always signaled by level crossings of a complex-extended Hamiltonian. We also find that integrability implies a reduced number of exceptional points. Both properties could uniquely characterize quantum integrability in small Hilbert spaces.

We introduce a method by which stochastic processes are mapped onto complex networks. As examples, we construct the networks for such time series as those for free-jet and low-temperature helium turbulence, the German stock market index (the DAX), and white noise. The networks are further studied by contrasting their geometrical properties, such as the mean length, diameter, clustering, and average number of connections per node. By comparing the network properties of the original time series investigated with those for the shuffled and surrogate series, we are able to quantify the effect of the long-range correlations and the fatness of the probability distribution functions of the series on the networks constructed. Most importantly, we demonstrate that the time series can be reconstructed with high precision by means of a simple random walk on their corresponding networks.

The graded reflection equation is investigated for the U_q[osp(r|2m)⁽¹⁾] vertex model. We have found diagonal solutions with at the most one free parameter and non-diagonal solutions with the number of free parameters depending on the number of bosonic (r) and fermionic (2m) degrees of freedom.

We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength h = HTk_B, as well as the Villain model on the square lattice. After a quench from a disordered initial state to T = 0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, L, we define and measure sample-dependent 'first passage time', t_r, which is the number of Monte Carlo steps until the energy is relaxed to the ground state value. The distribution of t_r, in particular its mean value, ⟨t_r(L)⟩, is shown to obey the scaling relation, ⟨t_r(L)⟩∼L²ln(L/L₀), for both models. Scaling of the autocorrelation function of the antiferromagnetic triangular model is shown to involve logarithmic corrections, both at H = 0 and at the field-induced Kosterlitz–Thouless transition: however, the autocorrelation exponent is found to be H-dependent.

Many empirical studies reveal that the weights and community structure are ubiquitous in various natural and artificial networks. In this paper, based on the SI disease model, we investigate the epidemic spreading in weighted scale-free networks with community structure. Two exponents, α and β, are introduced to weight the internal edges and external edges, respectively; and a tunable probability parameter q is also introduced to adjust the strength of community structure. We find the external weighting exponent β plays a much more important role in slackening the epidemic spreading and reducing the danger brought by the epidemic than the internal weighting exponent α. Moreover, a novel result we find is that the strong community structure is no longer helpful for slackening the danger brought by the epidemic in the weighted cases. In addition, we show the hierarchical dynamics of the epidemic spreading in the weighted scale-free networks with communities which is also displayed in the famous BA scale-free networks.

It has been shown that the communities of complex networks often overlap with each other. However, there is no effective method to quantify the overlapping community structure. In this paper, we propose a metric to address this problem. Instead of assuming that one node can only belong to one community, our metric assumes that a maximal clique only belongs to one community. In this way, the overlaps between communities are allowed. To identify the overlapping community structure, we construct a maximal clique network from the original network, and prove that the optimization of our metric on the original network is equivalent to the optimization of Newman's modularity on the maximal clique network. Thus the overlapping community structure can be identified through partitioning the maximal clique network using any modularity optimization method. The effectiveness of our metric is demonstrated by extensive tests on both artificial networks and real world networks with a known community structure. The application to the word association network also reproduces excellent results.

We introduce a new microcanonical dynamics for a large class of Ising systems isolated or maintained out of equilibrium by contact with thermostats at different temperatures. Such a dynamics is very general and can be used in a wide range of situations, including ones with disordered and topologically inhomogeneous systems. Focusing on the two-dimensional ferromagnetic case, we show that the equilibrium temperature is naturally defined, and that it can be consistently extended as a local temperature when far from equilibrium. This holds for homogeneous as well as for disordered systems. In particular, we will consider a system characterized by ferromagnetic random couplings . We show that the dynamics relaxes to steady states, and that heat transport can be described on average by means of a Fourier equation. The presence of disorder reduces the conductivity, the effect being especially appreciable for low temperatures. We finally discuss a possible singular behaviour arising for small disorder, i.e. in the limit .

We report the existence of enhanced heap stability as a result of the mixing of granular materials. Our setup consists of a rectangular container, filled with a binary mixture of granular matter up to some height h, that is rapidly opened at one wall to allow repose angle (θ_c) formation. We develop an empirical model for θ_c based on the Mohr–Coulomb failure criterion. The model is parameterized by an effective cohesion c and an effective coefficient of friction μ that depend on: (1) the granular proportions and (2) the c and μ for pure cases. Good agreement is achieved between the experiment and the model. We note that even the experimental fluctuations of θ_c as a function of granular proportions are well correlated (<2% deviation) with the computed uncertainty of the empirical model.

We consider the detection of correlated information sources in the ubiquitous code-division multiple-access (CDMA) scheme. We propose a message-passing based scheme for detecting correlated sources directly, with no need for source coding. The detection is done simultaneously over a block of transmitted binary symbols (word). Simulation results are provided, demonstrating a substantial improvement in bit error rate in comparison with the unmodified detector and the alternative of source compression. The robustness of the error-performance improvement is shown under practical model settings, including wrong estimation of the generating Markov transition matrix and finite-length spreading codes.

We consider bosons at Landau level filling ν = 1 on a thin torus. In analogy with previous work on fermions at filling ν = 1/2, we map the low-energy sector onto a spin-1/2 chain. While the fermionic system may realize the gapless XY phase, we show that typically this does not happen for the bosonic system. Instead, both delta function and Coulomb interaction lead to gapped phases in the bosonic system, and in particular we identify a phase corresponding to the non-Abelian Moore–Read state. In the spin language, the Hamiltonian is dominated by a ferromagnetic next-nearest-neighbor interaction, which leads to a description consistent with the non-trivial degeneracies of the ground and excited states of this phase of matter. In addition we comment on the similarities and differences of the two systems mentioned above and fermions at ν = 5/2.

Using their relationship with the free boson and the free symplectic fermion, we study the off-critical perturbations of SLE(4) and SLE(2) obtained by adding a mass term to the action. We compute the off-critical statistics of the source in the Loewner equation describing the two-dimensional interfaces. In these two cases we show that ratios of massive and massless partition functions, expressible as ratios of regularized determinants of massive and massless Laplacians, are (local) martingales for the massless interfaces. The off-critical drifts in the stochastic source of the Loewner equation are proportional to the logarithmic derivative of these ratios. We also show that massive correlation functions are (local) martingales for the massive interfaces. In the case of massive SLE(4), we use this property to prove a factorization of the free boson measure.

In contrast to the canonical case, microcanonical thermodynamic functions can show nonanalyticities also for finite systems. In this paper we contribute to the understanding of these nonanalyticities by working out the relation between nonanalyticities of the microcanonical entropy and its configurational counterpart. If the configurational microcanonical entropy ω_N^c(v) has a nonanalyticity at v = v_c, then the microcanonical entropy ω_N(ε) has a nonanalyticity at the same value ε = v_c of its argument for any finite value of the number of degrees of freedom N. The presence of the kinetic energy weakens the nonanalyticities such that, if the configurational entropy is p-times differentiable, the entropy is -times differentiable. In the thermodynamic limit, however, the behaviour is very different: the nonanalyticities no longer occur at the same values of the arguments, and the nonanalyticity of the microcanonical entropy is shifted to a larger energy. These results give a general explanation of the peculiar behaviour previously observed for the mean-field spherical model. With the hypercubic model we provide a further example illustrating our results.

We calculate a correlation function of the Jordan–Wigner operator in a class of free-fermion models formulated on an infinite one-dimensional lattice. We represent this function in terms of the determinant of an integrable Fredholm operator, convenient for analytic and numerical investigations. By using Wick's theorem, we avoid the form-factor summation customarily used in the literature for treating similar problems.

In this paper, we present an exactly solvable phase transition model in which the phase transition is purely statistically derived. The phase transition in this model is a generalized Bose–Einstein condensation. The exact expression for the thermodynamic quantity, which can be used to simultaneously describe both the gas phase and the condensed phase, is solved with the help of the homogeneous Riemann–Hilbert problem, so one can judge whether there exists a phase transition and determine the phase transition point mathematically rigorously. A generalized statistics in which the maximum occupation numbers of different quantum states can take on different values is introduced, as a generalization of Bose–Einstein and Fermi–Dirac statistics.

The scaling theory of Anderson localization is based on a global conductance g_L that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these two cases and analyze the statistics of the corresponding transmissions. We present detailed numerical results for the power-law random banded matrices (PRBM model). Our conclusions are: (i) in the presence of one isolated incoming wire and many outgoing wires, the transmission has the same multifractal statistics as the local density of states of the site where the incoming wire arrives and (ii) in the presence of backward scattering channels with respect to case (i), the statistics of the transmission is not multifractal anymore, but becomes monofractal. Finally, we also describe how these scattering geometries influence the statistics of the transmission off criticality.

We study the Sinai model for the diffusion of a particle in a one-dimensional random potential in the presence of a small concentration ρ of perfect absorbers using the asymptotically exact real space renormalization method. We compute the survival probability, the averaged diffusion front and return probability, the two-particle meeting probability, the distribution of total distance traveled before absorption and the averaged Green's function of the associated Schrödinger operator. Our work confirms some recent results of Texier and Hagendorf obtained by Dyson–Schmidt methods, and extends them to other observables and the presence of a drift. In particular the power law density of states is found to hold in all cases. Irrespective of the drift, the asymptotic rescaled diffusion front of surviving particles is found to be a symmetric step distribution, uniform for , where ξ(t) is a new length scale for survival ( in the absence of drift). Survival outside this sharp region is found to decay with a larger exponent, continuously varying with the rescaled distance x/ξ(t). A simple physical picture based on a saddle point is given, and universality is discussed.

We study the thermodynamic Casimir effect in thin films in the three-dimensional XY universality class. To this end, we simulate the improved two-component ϕ⁴ model on the simple cubic lattice. We use lattices up to the thickness L₀ = 33. On the basis of the results of our Monte Carlo simulations we compute the universal finite size scaling function θ that characterizes the behaviour of the thermodynamic Casimir force in the neighbourhood of the critical point. We confirm that leading corrections to the universal finite size scaling behaviour due to free boundary conditions can be expressed using an effective thickness L_0,eff = L₀+L_s, with L_s = 1.02(7). Our results are compared with experiments on films of ⁴He near the λ-transition, previous Monte Carlo simulations of the XY model on the simple cubic lattice and field-theoretic results. Our result for the finite size scaling function θ is essentially consistent with the experiments on films of ⁴He and the previous Monte Carlo simulations.

An extensive Monte Carlo study of the two-dimensional Ising model is made to investigate the statistical behavior of spin clusters and interfaces as a function of temperature, T. We use a tie-breaking rule to define interfaces of spin clusters on a square lattice with strip geometry and show that such definitions are consistent with conformally invariant properties of interfaces at the critical temperature, T_c. The effective fractal dimensions of spin clusters and interfaces (d_c and d_I, respectively) are obtained as a function of temperature. We find that the effective fractal dimension of the spin clusters varies almost linearly with temperature in three different regimes. It is also found that the effective fractal dimension of the interfaces undergoes a sharp crossover around T_c, between the values 1 and 1.75 at low and high temperatures, respectively. We also check the finite-size scaling hypothesis for the percolation probability, and the average mass of the largest spin cluster is in a good agreement with the theoretical predictions.

Using the recently derived dissipation theorem and a corollary of the transient fluctuation theorem (TFT), namely the second-law inequality, we derive the unique time independent, equilibrium phase space distribution function for an ergodic Hamiltonian system in contact with a remote heat bath. We prove under very general conditions that any deviation from this equilibrium distribution breaks the time independence of the distribution. Provided temporal correlations decay, we show that any nonequilibrium distribution that is an even function of the momenta eventually relaxes (not necessarily monotonically) to the equilibrium distribution. Finally we prove that the negative logarithm of the microscopic partition function is equal to the thermodynamic Helmholtz free energy divided by the thermodynamic temperature and Boltzmann's constant. Our results complement and extend the findings of modern ergodic theory and show the importance of dissipation in the process of relaxation towards equilibrium.

Correlations are known to play a crucial role in determining the structure of complex networks. Here we study how their presence affects the computation of the percolation threshold in random hypergraphs. In order to mimic the correlation in real networks, we build hypergraphs from generalized hidden variable ensembles and we study the percolation transition by mapping this problem to the fully connected Potts model with heterogeneous couplings.

A generalized definition of average, termed the q-average, is widely employed in the field of nonextensive statistical mechanics. Recently, it has however been pointed out that such an average value may behave unphysically under specific deformations of probability distributions. Here, the following three issues are discussed and clarified. Firstly, the deformations considered are physical and may be realized experimentally. Secondly, in view of the thermostatistics, the q-average is unstable in both finite and infinite discrete systems. Thirdly, a naive generalization of the discussion to continuous systems misses a point, and a norm better than the L¹-norm should be employed for measuring the distance between two probability distributions. Consequently, stability of the q-average is shown not to be established in all of the cases.

Inferring the sequence of states from observations is one of the most fundamental problems in hidden Markov models. In statistical physics language, this problem is equivalent to computing the marginals of a one-dimensional model with a random external field. While this task can be accomplished through transfer matrix methods, it becomes quickly intractable when the underlying state space is large.

This paper develops several low complexity approximate algorithms to address this inference problem when the state space becomes large. The new algorithms are based on various mean-field approximations of the transfer matrix. Their performances are studied in detail on a simple realistic model for DNA pyrosequencing.

This work is concerned with the quasi-classical limit of the boundary quantum inverse scattering method for the twisted sl(2|1)⁽²⁾ vertex model with diagonal K-matrices. In this limit Gaudin's Hamiltonians with diagonal boundary terms are presented and diagonalized.

We discuss the well known Einstein and the Kubo fluctuation-dissipation relations (FDRs) in the wider framework of a generalized FDR for systems with a stationary probability distribution. A multivariate linear Langevin model, which includes dynamics with memory, is used as a treatable example to show how the usual relations are recovered only in particular cases. This study brings to the fore the ambiguities of a check of the FDR done without knowing the significant degrees of freedom and their coupling. An analogous scenario emerges in the dynamics of diluted shaken granular media. There, the correlation between position and velocity of particles, due to spatial inhomogeneities, induces violation of usual FDRs. The search for the appropriate correlation function which could restore the FDR can be more insightful than a definition of 'non-equilibrium' or 'effective temperatures'.

The emergence of cooperation in self-centered individuals has been a major puzzle in the study of evolutionary ethics. Reciprocal altruism is one of the explanations put forward and the prisoner's dilemma has been a paradigm in this context. The emergence of cooperation was demonstrated for the prisoner's dilemma on a lattice with synchronous update. However, the cooperation disappeared for asynchronous update and the general validity of the conclusions was questioned. Neither synchronous nor asynchronous updates are realistic for natural systems. In this paper, we make a detailed study of a more realistic system of semi-synchronous updates where pN agents are updated at every time instant. We observe a transition from an all-defector state to a mixed state as a function of p. Our studies indicate that despite it being a transition from an absorbing state, it is a first-order transition. Furthermore, we used a damage spreading technique to demonstrate that the transition in this system could be classified as a frozen–chaotic transition.

Boltzmann–Gibbs measures generated from logarithmically correlated random potentials are multifractal. We investigate the abrupt change ('pre-freezing') of multifractality exponents extracted from the averaged moments of the measure—the so-called inverse participation ratios. The pre-freezing can be identified with termination of the disorder-averaged multifractality spectrum. The naive replica limit employed to study a one-dimensional variant of the model is shown to break down at the pre-freezing point. Further insights are possible when employing zero-dimensional and infinite-dimensional versions of the problem. In particular, the latter version allows one to identify the pattern of the replica symmetry breaking responsible for the pre-freezing phenomenon.

The conserved Manna model with a planar absorbing boundary is studied in various space dimensions. We present a heuristic argument that allows one to compute the surface-critical exponent in one dimension analytically. Moreover, we discuss the mean-field limit that is expected to be valid in d>4 space dimensions and demonstrate how the corresponding partial differential equations can be solved.

The dynamics of linear stochastic growth equations on growing substrates is studied. The substrate is assumed to grow in time following the power law t^γ, where the growth index γ is an arbitrary positive number. Two different regimes are clearly identified: for small γ the interface becomes correlated, and the dynamics is dominated by diffusion; for large γ the interface stays uncorrelated, and the dynamics is dominated by dilution. In this second regime, for short time intervals and spatial scales the critical exponents corresponding to the non-growing substrate situation are recovered. For long time differences or large spatial scales the situation is different. Large spatial scales show the uncorrelated character of the growing interface. Long time intervals are studied by means of the auto-correlation and persistence exponents. It becomes apparent that dilution is the mechanism by which correlations are propagated in this second case.

We study a quantum double model whose degrees of freedom are Ising anyons. The terms of the Hamiltonian of this system give rise to a competition between single and double topologies. By studying the energy spectra of the Hamiltonian at different values of the coupling constants, we find extended gapless regions which include a large number of critical points described by conformal field theories with central charge c = 1. These theories are part of the orbifold of the bosonic theory compactified on a circle. We observe that the Hilbert space of our anyonic model can be associated with extended Dynkin diagrams of affine Lie algebras, which yields exact solutions at some critical points. In certain special regimes, our model corresponds to the Hamiltonian limit of the Ashkin–Teller model, and hence integrability over a wide range of coupling parameters is established.

We study the steady state behavior of a confined quantum Brownian particle subjected to a space dependent, rapidly oscillating time-periodic force. To leading order in the period of driving, the result of the oscillating force is an effective static potential which has a quantum dissipative contribution, V_QD, which adds on to the classical result. This is shown using a coherent state representation of bath oscillators. V_QD is evaluated exactly in the case of an Ohmic dissipation bath. It is strongest for intermediate values of the damping, where it can have pronounced effects.

The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points.

Perpetual American options are financial instruments that can be readily exercised and do not mature. In this paper we study in detail the problem of pricing this kind of derivatives, for the most popular flavour, within a framework in which some of the properties—volatility and dividend policy—of the underlying stock can change at a random instant of time but in such a way that we can forecast their final values. Under this assumption we can model actual market conditions because most relevant facts usually entail sharp predictable consequences. The effect of this potential risk on perpetual American vanilla options is remarkable: the very equation that will determine the fair price depends on the solution to be found. Sound results are found under the optics both of finance and physics. In particular, a parallelism among the overall outcome of this problem and a phase transition is established.

We use dynamic light scattering and numerical simulations to study the approach to equilibrium and the equilibrium dynamics of systems of colloidal hard spheres over a broad range of densities, from dilute systems up to very concentrated suspensions undergoing glassy dynamics. We discuss several experimental issues (sedimentation, thermal control, non-equilibrium ageing effects, dynamic heterogeneity) arising when very large relaxation times are measured. When analyzed over more than seven decades in time, we find that the equilibrium relaxation time, τ_α, of our system is described by the algebraic divergence predicted by mode-coupling theory over a window of about three decades. At higher density, τ_α increases exponentially with distance to a critical volume fraction φ₀, which is much larger than the mode-coupling singularity. This is reminiscent of the behavior of molecular glass-formers in the activated regime. We compare these results to previous work, carefully discussing crystallization and size polydispersity effects. Our results suggest the absence of a genuine algebraic divergence of τ_α in colloidal hard spheres.

We use rigorous arguments and Monte Carlo simulations to study the thermodynamics and the topological properties of self-avoiding walks on the cubic lattice subjected to an external force f. The walks are anchored at one or both endpoints to an impenetrable plane at Z = 0 and the force is applied in the Z-direction. If a force is applied to the free endpoint of an anchored walk, then a model of pulled walks is obtained. If the walk is confined to a slab and a force is applied to the top bounding plane, then a model of stretched walks is obtained. For both models we prove the existence of the limiting free energy for any value of the force and we show that, for compressive forces, the thermodynamic properties of the two models differ substantially. For pulled walks we prove the existence of a phase transition that, by numerical simulation, we estimate to be second order and located at f = 0. By using a pattern theorem for large positive forces we show that almost all sufficiently long stretched walks are knotted. We examine the entanglement complexity of stretched and pulled walks; our numerical results show a sharp reduction with increasing pulling and stretching forces. Finally, we also examine models of pulled and stretched loops. We prove the existence of limiting free energies in these models and consider the knot probability numerically as a function of the applied pulling or stretching force.

For systems in an externally controllable time dependent potential, the optimal protocol minimizes the mean work spent in a finite time transition between given initial and final values of a control parameter. For an initially thermalized ensemble, we consider both Hamiltonian evolution for classical systems and Schrödinger evolution for quantum systems. In both cases, we show that for harmonic potentials, the optimal work is given by the adiabatic work even in the limit of short transition times. This result is counter-intuitive because the adiabatic work is substantially smaller than the work for an instantaneous jump. We also perform numerical calculations for the optimal protocol for Hamiltonian dynamics in an anharmonic quartic potential. For a two-level spin system, we give examples where the adiabatic work can be reached in either a finite or an arbitrarily short transition time depending on the allowed parameter space.

We consider a smooth, rotationally invariant, centered Gaussian process in the plane, with arbitrary correlation matrix C_tt'. We study the winding angle ϕ_t, around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrix C_tt'. For most stationary processes C_tt' = C(t−t') the winding angle exhibits diffusion at large time with diffusion coefficient . Correlations of exp(inϕ_t) with integer n, the distribution of the angular velocity , and the variance of the algebraic area are also obtained. For smooth processes with stationary increments (random walks) the variance of the winding angle grows as , with proper generalizations to the various classes of fractional Brownian motion. These results are tested numerically. Non-integer n is studied numerically.

We consider the Jaynes–Cummings model of a single quantum spin s coupled to a harmonic oscillator in a parameter regime where the underlying classical dynamics exhibits an unstable equilibrium point. This state of the model is relevant to the physics of cold atom systems, in non-equilibrium situations obtained by fast sweeping through a Feshbach resonance. We show that in this integrable system with two degrees of freedom, for any initial condition close to the unstable point, the classical dynamics is controlled by a singularity of the focus–focus type. In particular, it displays the expected monodromy, which forbids the existence of global action-angle coordinates. Explicit calculations of the joint spectrum of conserved quantities reveal the monodromy at the quantum level, as a dislocation in the lattice of eigenvalues. We perform a detailed semi-classical analysis of the associated eigenstates. Whereas most of the levels are well described by the usual Bohr–Sommerfeld quantization rules, properly adapted to polar coordinates, we show how these rules are modified in the vicinity of the critical level. The spectral decomposition of the classically unstable state is computed, and is found to be dominated by the critical WKB states. This provides a useful tool with which to analyze the quantum dynamics starting from this particular state, which exhibits an aperiodic sequence of solitonic pulses with a rather well defined characteristic frequency.

A non-equilibrium particle transport model, the totally asymmetric exclusion process, is studied on a one-dimensional lattice with a hierarchy of fixed long range connections. This model breaks the particle–hole symmetry observed on an ordinary one-dimensional lattice and results in a surprisingly simple phase diagram, without a maximum current phase. Numerical simulations of the model with open boundary conditions reveal a number of dynamic features and suggest possible applications.

We use a discrete model to study the non-equilibrium dynamics of a slowly driven elastic string in a two-dimensional disordered medium at finite temperatures. We focus on the local activity statistics to show that it can be related to global observables like the average interface velocity and the temporal correlations of the velocity fluctuations. For low temperatures the string exhibits typical creep motion and the activity statistics follows a power law, consistent with an exponential distribution of energy barriers. However, we find that the activity statistics is essentially different when the temperature is low enough, suggesting a different relaxation mechanism as . We argue that this is due to the generic non-equilibrium nature of our model in the absence of thermal fluctuations.

In this paper the question of statistical properties of block-hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by a 'mipmapping' procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined via a hierarchy of the Bernoulli distributions {q₁,q₂,...} on matrix elements, where q_γ depends on the hierarchy level γ as q_γ = p^−μγ (μ>0). For the spectral density we clearly see scale-free behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances σ_γ = p^−νγ, the tail of the spectral density, ρ_G(λ), behaves as ρ_G(λ)∼|λ|^{−(2−ν)/(1−ν)} for and 0<ν<1, while for ν≥1 the power-law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly-scale fractal behavior extended over a very broad range of scales.

Spatially explicit models are widely used in today's mathematical ecology and epidemiology to study persistence and extinction of populations as well as their spatial patterns. Here we extend the earlier work on static dispersal between neighboring individuals to the mobility of individuals as well as multi-patch environments. As is commonly found, the basic reproductive ratio is maximized for the evolutionarily stable strategy for disease persistence in mean field theory. This has important implications, as it implies that for a wide range of parameters the infection rate tends to a maximum. This is opposite to the present result obtained from spatially explicit models, which is that the infection rate is limited by an upper bound. We observe the emergence of trade-offs of extinction and persistence for the parameters of the infection period and infection rate, and show the extinction time as having a linear relationship with respect to system size. We further find that higher mobility can pronouncedly promote the persistence of the spread of epidemics, i.e., a phase transition occurs from the extinction domain to the persistence domain, and the wavelength of the spirals increases with the mobility ratio enhancement and will ultimately saturate at a certain value. Furthermore, for the multi-patch case, we find that lower coupling strength leads to anti-phase oscillation of the infected fraction, while higher coupling strength corresponds to in-phase oscillation.

Data clustering, including problems such as finding network communities, can be put into a systematic framework by means of a Bayesian approach. Here we address the Bayesian formulation of the problem of finding hypergraph communities. We start by introducing a hypergraph generative model with a built-in group structure. Using a variational calculation we derive a variational Bayes algorithm, a generalized version of the expectation maximization algorithm with a built-in penalization for model complexity or bias. We demonstrate the good performance of the variational Bayes algorithm using test examples, including finding network communities. A MATLAB code implementing this algorithm is provided as supplementary material.

The ensemble of random Markov matrices is introduced as a set of Markov or stochastic matrices with the maximal Shannon entropy. The statistical properties of the stationary distribution π, the average entropy growth rate h and the second-largest eigenvalue ν across the ensemble are studied. It is shown and heuristically proven that the entropy growth rate and second-largest eigenvalue of Markov matrices scale on average with the dimension of the matrices d as h∼log(O(d)) and |ν|∼d^−1/2, respectively, yielding the asymptotic relation hτ_c∼1/2 between the entropy h and the correlation decay time τ = −1/log|ν|. Additionally, the correlation between h and τ_c is analysed; it decreases with increasing dimension d.

We study a general model of a granular Brownian ratchet consisting of an asymmetric object moving on a line and surrounded by a two-dimensional granular gas, which in turn is coupled to an external random driving force. We discuss the two resulting Boltzmann equations describing the gas and the object in the dilute limit and obtain a closed system for the first few moments of the system velocity distributions. Predictions for the net ratchet drift, the variance of its velocity fluctuations and the transition rates in the Markovian limit are compared to numerical simulations and a fair agreement is observed.

We solve a random energy model with complex replica number and complex temperature values, and discuss the ensuing phase structure. A connection with string models and their phase structure is analyzed from the REM point of view. The REM analysis yields a few integer dimensions as special points of the REM phase diagram. For N = 1 superstrings, there is a distinguished dimension 5.

Finding the optimal assignment in budget-constrained auctions is a combinatorial optimization problem with many important applications, a notable example being in the sale of advertisement space by search engines (in this context the problem is often referred to as the off-line AdWords problem). On the basis of the cavity method of statistical mechanics, we introduce a message-passing algorithm that is capable of solving efficiently random instances of the problem extracted from a natural distribution, and we derive from its properties the phase diagram of the problem. As the control parameter (average value of the budgets) is varied, we find two phase transitions delimiting a region in which long-range correlations arise.

We study mappings between different classical spin systems that leave the partition function invariant. As recently shown in Van den Nest et al (2008 Phys. Rev. Lett. 100 110501), the partition function of the 2D square lattice Ising model in the presence of an inhomogeneous magnetic field can specialize to the partition function of any Ising system on an arbitrary graph. In this sense the 2D Ising model is said to be 'complete'. However, in order to obtain the above result, the coupling strengths on the 2D lattice must assume complex values, and thus do not allow for a physical interpretation. Here we show how a complete model with real—and, hence, 'physical'—couplings can be obtained if the 3D Ising model is considered. We furthermore show how to map general q-state systems with possibly many-body interactions to the 2D Ising model with complex parameters, and give completeness results for these models with real parameters. We also demonstrate that the computational overhead in these constructions is in all relevant cases polynomial. These results are proved by invoking a recently found cross-connection between statistical mechanics and quantum information theory, where partition functions are expressed as quantum mechanical amplitudes. Within this framework, there exists a natural correspondence between many-body quantum states that allow for universal quantum computation via local measurements only, and complete classical spin systems.

Commentary on `Asymptotic statistics of the n-sided planar Poisson–Voronoi cell: II. Heuristics', by H J Hilhorst, 2009 J. Stat. Mech.P05007.