Thermodynamical Limit for Correlated Gaussian Random Energy Models

 Let {E _Σ (N)} _ΣΣN be a family of |Σ _N |=2 ^N centered unit Gaussian random variables defined by the covariance matrix C _N of elements c _N (Σ,τ):=Av(E _Σ (N)E _τ (N)) and the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N=N ₁ +N ₂ , and all pairs (Σ,τ)Σ _N ×Σ _N :

Neighborhood preferences in random matching problems

We consider a class of random matching problems where the distance between two points has a probability law which, for a small distance l, goes like l^r. In the framework of the cavity method, in the limit of an infinite number of points, we derive equations for p_k, the probability for some given point to be matched to its kth nearest neighbor in the optimal configuration. These equations are solved in two limiting cases: r = 0 -- where we recover p _k = 1/2^k, as numerically conjectured by Houdayer et al. and recently rigorously proved by Aldous -- and r→ + ∞. For 0 < r < + ∞, we are not able to solve the equations analytically, but we compute the leading behavior of p_k for large k. Received 14 February 2001     

Mean square number fluctuation for a fermion source and its dependence on neutrino mass for the universal cosmic neutrino background

Using the general formulation for obtaining chemical potentialμ of an ideal Fermi gas of particles at temperature T, with particle rest mass m₀ and average density 〈N〉/V, the dependence of the mean square number fluctuation 〈ΔN ²〉/V on the particle mass m₀ has been calculated explicitly. The numerical calculations are exact in all cases whether rest mass energym ₀c² is very large (non-relativistic case), very small (ultra-relativistic case) or of the same order as the thermal energy k_BT. Application of our results to the detection of the universal very low energy cosmic neutrino background (CNB), from any of the three species of neutrinos, shows that it is possible to estimate the neutrino mass of these species if from approximate experimental measurements of their momentum distribution one can extract, someday, not only the density 〈N _v〉/V but also the mean square fluctuation 〈Δ _v ² 〉/V. If at the present epoch, the universe is expanding much faster than thermalization rate for CNB, it is shown that our analysis leads to a scaled neutrino massm _v instead of the actual massm _0v .     

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions , with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p _N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p _N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k ^th moment of p _N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p _N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.     

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001     

Modeling DNA and Virus Trafficking in the Cell Cytoplasm

Cytosol trafficking is a limiting step of viral infection or DNA delivery. Starting from the cell surface, most viruses have to travel through a crowded and risky environment in order to reach a small nuclear pore. This work is dedicated to estimating the probability p _N of a viral arrival success and, in that case, the mean time τ _N it takes. Viral movement is described by a stochastic equation, containing both a drift and a Brownian component. The drift part represents the movement along microtubules, while the Brownian component corresponds to the free diffusion. The success of a viral infection is limited by a killing activity occurring inside the cytoplasm. We model the killing activity by a steady state killing rate k. Because nuclear pores occupy a small fraction of the nuclear area, we use this property to obtain asymptotic estimates of p _N and τ _N as a function of the diffusion constant D, the amplitude of the drift B and the killing rate k. This paper is dedicated to my wife Nathalie Rouach. D. H. is supported by the program “Chaire d’Excellence.” D.H is incumbent to the Haas Russell Chair     

Dimensional Crossover of Magnetic Transition in Diluted Ising Spin Nets Probed by Muon Spin Relaxation

The magnetic ordering process of Ising spins on diluted square lattice was studied by muon spin relaxation using model compounds Rb₂Co _c Mg_1−c F₄. Muon relaxation shows an anomaly at a remarkably higher temperature T _N ^μSR than the transition temperature determined by neutron Bragg scattering T _N ^ND near the percolation threshold for square lattice (c _p=0.593). The difference between the two temperatures amounts to 50% of T _N ^ND just above c _p. The field cooling effect of DC magnetic susceptibility is appreciable below T _N ^ND while the temperature of the anomaly in AC susceptibility approaches to T _N ^μSR as the frequency is increased. It was concluded that there is a crossover from two-dimensional ordering at T _N ^μSR to three-dimensional ordering at T _N ^ND but the two-dimensional order between T _N ^μSR and T _N ^ND has slow fluctuations due to the fractal structure with a plenty of weak links. This revised version was published online in September 2006 with corrections to the Cover Date.     

The Critical Point of k-Clique Percolation in the Erdős–Rényi Graph

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdős–Rényi graph. When the probability p of two nodes being connected is above a certain threshold p _c (k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p _c (k).     

PHOBOS at RHIC: Some global observations

Particle production in Au+Au collisions has been measured in the PHOBOS experiment at RHIC for a range of collision energies for a large span of pseudorapidities, |η| < 5.4. Three empirical observations have emerged from this data set which require theoretical examination. First, there is clear evidence of limiting fragmentation. Namely, particle production in central Au + Au collisions, when expressed as dN/dη′ ( η′ ≡ – y_beam), becomes energy independent at high energy for a broad region of η′ around η′ = 0. This energy-independent region grows with energy, allowing only a limited region (if any) of longitudinal boost-invariance. Second, there is a striking similarity between particle production in e⁺e⁻and Au + Au collisions (scaled by the number of participating nucleon pairs). Both the total number of produced particles and the longitudinal distribution of produced particles are approximately the same in e⁺e⁻and in scaled Au + Au. This observation This presentation is based in large part on the PHOBOS summary talk by M Baker at the16th Int. Conf. on Ultrarelativistic Nucleus- Nucleus Collisions, Quark Matter 2002, Nantes, France was not predicted and has not been explained. Finally, particle production has been found to scale approximately with the number of participating nucleon pairs for (N _part ) > 65. This scaling occurs both for the total multiplicity and for highp _T particles (3 <p _T < 4.5 GeV/c). This presentation is based in large part on the PHOBOS summary talk by M Baker at the16th Int. Conf. on Ultrarelativistic Nucleus-Nucleus Collisions, Quark Matter 2002, Nantes, France     

Bootstrap Percolation on Homogeneous Trees Has 2 Phase Transitions

We study the threshold θ bootstrap percolation model on the homogeneous tree with degree b+1, 2≤θ≤b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call p _f , such that a) for p>p _f , the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p<p _f , then for almost every initial configuration, the final bootstrapped configuration has density of occupied vertices less than 1. In this paper, we establish the existence of a distinct critical value for p, p _c , such that 0<p _c <p _f , with the following properties: 1) if p≤p _c , then for almost every initial configuration there is no infinite cluster of occupied vertices in the final bootstrapped configuration; 2) if p>p _c , then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p<p _c , the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail; 4) at p=p _c , the expected occupied cluster size in the final bootstrapped configuration is infinite; 5) the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on [0,p _f ] and analytic on (p _c ,p _f ), admitting an analytic continuation from the right at p _c and, only in the case θ=b, also from the left at p _f . L.R.G. Fontes partially supported by the Brazilians CNPq through grants 475833/2003-1, 307978/2004-4 and 484351/2006-0, and FAPESP through grant 04/07276-2. R.H. Schonmann partially supported by the American N.S.F. through grant DMS-0300672.     

Perturbations in the rotational structure of the 4<Emphasis Type="Italic">p</Emphasis> complex of H<Subscript>2</Subscript> molecule terms

A statistical analysis of all the available data on the wave numbers of spectral lines related to triplet-triplet electronic-vibrational-rotational (rovibronic) radiation transitions into the H₂ molecule (1sσ2sσ) a ³Σ _g ⁺ electronic state was performed for the first time. This allowed us to check and refine the controversial identification of several spectral lines. Optimum rovibronic term values were found for 15 electronic states, including the (4pσ)f ³Σ _u ⁺, (4pπ)k ³Π _u ⁺, and (4pπ)k ³Π _u ⁻ states studied in this work. The ratios between the oscillator strengths of R- and P-branch lines with common upper levels (branching coefficients) for the f ³Σ _u ⁺ → a ³Σ _g ⁺ and k ³Π _u ⁺ → a ³Σ _g ⁺ systems of H₂ molecule bands were measured for the first time. Substantial deviations of the measured branching coefficients from the corresponding ratios between the Henl-London factors were observed. The deviations monotonically increased as the rotational quantum number N grew, which, in combination with substantial Λ-doubling in the k ³Π _u state, was evidence of an important role played by electronicrotational interaction in the 4pσ³Σ _u ⁺ and 4pπ³Π _u ⁺ adiabatic electronic states. A strong correlation was observed between the N dependences of branching coefficients for transitions from the mutually perturbed f ³Σ _u ⁺ and k ³Π _u ⁺ electronic states. The results of this work show that the measured branching coefficients are a much more sensitive and capacious channel of information about perturbation effects than rovibronic term values.     

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ³ for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6π ν j where j is the current density of the particles, and of a friction term 6π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ε=1/N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure , where x _k is the center of the k-th ball and v _k its instantaneous velocity. This can be seen as a generalization of Allaire’s result in [Arch. Ration. Mech. Anal. 113:209–259, [1991]] who considered the case of periodically distributed x _k s with v _k =0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays.     

Quantum oscillations of the resistivity and hall coefficient and the quantum limit in Bi<Subscript>0.93</Subscript>Sb<Subscript>0.07</Subscript> alloys in a magnetic field along the trigonal axis

The dependences of the electrical resistivity ρ and the Hall coefficient R on the magnetic field have been measured for single-crystal samples of the n-Bi_0.93Sb_0.07 semiconductor alloys with electron concentrations in the range 1 × 10¹⁶ cm⁻³ < n < 2 × 10¹⁸ cm⁻³. It has been found that the measured dependences exhibit Shubnikov-de Haas quantum oscillations. The magnetic fields corresponding to the maxima of the quantum oscillations of the electrical resistivity are in good agreement with the calculated values of the magnetic fields in which the Landau quantum level with the number N intersects the Fermi level. The quantum oscillations of the Hall coefficient with small numbers are characterized by a significant spin splitting. In a magnetic field directed along the trigonal axis, the quantum oscillations of the resistivity ρ and the Hall coefficient R are associated with electrons of the three-valley semiconductor and are in phase with the magnetic field. In the case of a magnetic field directed parallel to the binary axis, the quantum oscillations associated both with electrons of the secondary ellipsoids in weaker magnetic fields and with electrons of the main ellipsoid in strong magnetic fields (after the overflow of electrons from the secondary ellipsoids to the main ellipsoid) are also in phase. In magnetic fields of the quantum limit ħω _c /2 ≥ E _F, the electrical conductivity increases with an increase in the magnetic field: σ₂₂(H) ∼ H ^k . A theoretical evaluation of the exponent in this expression for a nonparabolic semiconductor leads to values of k close to the experimental values in the range 4 ≤ k ≤ 4.6, which were obtained for samples of the semiconductor alloys with different electron concentrations. A further increase in the magnetic field results in a decrease of the exponent k and in the transition to the inequality σ₂₂(H) ≤ σ₂₁(H).     

Finite-dimensional representations of a shock algebra

Abstruct The algebra describing a shock measure in the asymmetric simple exclusion model, seen from a second class particle, has finite-dimensional representations if and only if the asymmetry parameterp of the model and the left and right asymptotic densitiesp _± of the shock satisfy [(1−p)/p] ^r =p ₋(1−p ₊)/p ₊(1−p ₋) for some integerr≥1; the minimal dimension of the representation is then 2r. These representations can be used to calculate correlation functions in the model.     

Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice

The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameterk is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term,d _n,1 (k) is expressed by Gauss' hypergeometric series with a variablek. Since the ADBP includes the ordinary directed bond percolation as a special case withk=1, our results give another proof for the Baxter-Guttmann's conjecture thatd _n,1(1) is given by the Catalan number, which was recently proved by Bousquet-Mélou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis byDlog Padé approximations suggests that the critical value depends onk, while asymmetry does not change the critical exponent of percolation probability.     

Swelling of two-dimensional polymer rings by trapped particles

The mean area of a two-dimensional Gaussian ring of N monomers is known to diverge when the ring is subject to a critical pressure differential, p _c ∼ N ^-1. In a recent publication (Eur. Phys. J. E 19, 461 (2006)) we have shown that for an inextensible freely jointed ring this divergence turns into a second-order transition from a crumpled state, where the mean area scales as 〈A〉 ∼ N, to a smooth state with 〈A〉 ∼ N ². In the current work we extend these two models to the case where the swelling of the ring is caused by trapped ideal-gas particles. The Gaussian model is solved exactly, and the freely jointed one is treated using a Flory argument, mean-field theory, and Monte Carlo simulations. For a fixed number Q of trapped particles the criticality disappears in both models through an unusual mechanism, arising from the absence of an area constraint. In the Gaussian case the ring swells to such a mean area, 〈A〉 ∼ NQ, that the pressure exerted by the particles is at p _c for any Q. In the freely jointed model the mean area is such that the particle pressure is always higher than p _c, and 〈A〉 consequently follows a single scaling law, 〈A〉 ∼ N ² f (Q/N), for any Q. By contrast, when the particles are in contact with a reservoir of fixed chemical potential, the criticality is retained. Thus, the two ensembles are manifestly inequivalent in these systems. An erratum to this article is available at .     

Smoothening transition of a two-dimensional pressurized polymer ring

We revisit the problem of a two-dimensional polymer ring subject to an inflating pressure differential. The ring is modeled as a freely jointed closed chain of N monomers. Using a Flory argument, mean-field calculation and Monte Carlo simulations, we show that at a critical pressure, p_c ∼ N^-1, the ring undergoes a second-order phase transition from a crumpled, random-walk state, where its mean area scales as 〈A〉 ∼ N, to a smooth state with 〈A〉 ∼ N². The transition belongs to the mean-field universality class. At the critical point a new state of polymer statistics is found, in which 〈A〉 ∼ N^3/2. For p ≫ p_c we use a transfer-matrix calculation to derive exact expressions for the properties of the smooth state.     

The Scaling Limit Geometry of Near-Critical 2D Percolation

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = p _c+λδ^1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = p _c, based on SLE ₆. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of “macroscopically pivotal” lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.     

Continuity of percolation probability on hyperbolic graphs

LetT _k be a forwarding tree of degreek where each vertex other than the origin hask children and one parent and the origin hask children but no parent (k2). DefineG to be the graph obtained by adding toT _k nearest neighbor bonds connecting the vertices which are in the same generation.G is regarded as a discretization of the hyperbolic planeH ² in the same sense thatZ ^d is a discretization ofR ^d . Independent percolation onG has been proved to have multiple phase transitions. We prove that the percolation probabilityO(p) is continuous on [0,1] as a function ofp.     

Spin interference in the spin density matrix of a compound nucleus in reactions of nuclear fission by cold polarized neutrons

Spin density matrices of neutron resonance states of a compound nucleus formed in the reaction of capture of a polarized neutron by a non-oriented target nucleus for different directions of neutron polarization vector are constructed within the quantum fission theory. The obtained spin matrices are used to calculate T-odd asymmetries in differential cross sections of ternary nuclear fission with the emission of different third particles. It is demonstrated that the expressions for T-odd asymmetries in the cases of neutron polarization direction [(p)\vec]_n\vec p_n along the x and y axes in the laboratory reference frame differ by the values of the unified correlator of the form ( [(p)\vec]_n ,[ [(k)\vec]_LF ,[(k)\vec]₃ ] )\left( {\vec p_n ,\left[ {\vec k_{LF} ,\vec k_3 } \right]} \right) (where [(k)\vec]_LF\vec k_{LF} and [(k)\vec]₃\vec k_3 are the wave vectors of a light fission fragment and the third particle, respectively), and are transformed into one another if the laboratory reference frame in which [(p)\vec]_n\vec p_n is directed along the x axis is rotated to a laboratory reference frame in which [(p)\vec]_n\vec p_n is directed along the y axis. It is shown that T-odd TRI and ROT asymmetries are associated, respectively, with the odd and even components of the amplitudes of the angular distribution of third particles perturbed by the collective rotation of a polarized fissile nucleus, and each of these amplitudes can be considerably amplified (or suppressed) relative to one another due to the interference from fission amplitudes of pairs of neutron resonances sJ _s and s′J _s′ .