Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

We consider the ensemble of adjacency matrices of Erd?s-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption \({p N \gg N^{2/3}}\), we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd?s-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd?s-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.     

1 / <Emphasis Type="Italic">n</Emphasis> Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy

Using a 1 / n expansion, that is an expansion in descending powers of n, for the number of matchings in regular graphs with 2n vertices, we study the monomer-dimer entropy for two classes of graphs. We study the difference between the extensive monomer-dimer entropy of a random r-regular graph G (bipartite or not) with 2n vertices and the average extensive entropy of r-regular graphs with 2n vertices, in the limit \(n \rightarrow \infty \). We find a series expansion for it in the numbers of cycles; with probability 1 it converges for dimer density \(p < 1\) and, for G bipartite, it diverges as \(|\mathrm{ln}(1-p)|\) for \(p \rightarrow 1\). In the case of regular lattices, we similarly expand the difference between the specific monomer-dimer entropy on a lattice and the one on the Bethe lattice; we write down its Taylor expansion in powers of p through the order 10, expressed in terms of the number of totally reducible walks which are not tree-like. We prove through order 6 that its expansion coefficients in powers of p are non-negative.     

The Condensation Phase Transition in Random Graph Coloring

Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of interactions is induced by a sparse random graph or hypergraph. One example of such a model is the graph coloring problem on the Erd?s–Renyi random graph G(n, d/n), which can be viewed as the zero temperature case of the Potts antiferromagnet. The cavity method predicts that in addition to the k-colorability phase transition studied intensively in combinatorics, there exists a second phase transition called the condensation phase transition (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In fact, there is a conjecture as to the precise location of this phase transition in terms of a certain distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k₀.     

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

The Rényi entropies R_p [ ρ], p> 0,≠ 1 of the highly-excited quantum states of the D-dimensional isotropicharmonic oscillator are analytically determined by use of the strong asymptotics of theorthogonal polynomials which control the wavefunctions of these states, the Laguerrepolynomials. This Rydberg energetic region is where the transition from classical toquantum correspondence takes place. We first realize that these entropies are closelyconnected to the entropic moments of the quantum-mechanical probability ρ_n(r)density of the Rydberg wavefunctions Ψ_{n,l, { μ}}(r); so, to the?_p-norms of the associated Laguerrepolynomials. Then, we determine the asymptotics n → ∞ of these norms by use of modern techniques ofapproximation theory based on the strong Laguerre asymptotics. Finally, we determine thedominant term of the Rényi entropies of the Rydberg states explicitly in terms of thehyperquantum numbers (n,l), the parameter order p and the universedimensionality D for all possible cases D ≥ 1. We find that (a) theRényi entropy power decreases monotonically as the order p is increasing and (b) thedisequilibrium (closely related to the second order Rényi entropy), which quantifies theseparation of the electron distribution from equiprobability, has a quasi-Gaussianbehavior in terms of D.     

Models of random graph hierarchies

We introduce two models of inclusion hierarchies: random graph hierarchy (RGH) and limited random graph hierarchy (LRGH). In both models a set of nodes at a given hierarchy level is connected randomly, as in the Erd?s-Rényi random graph, with a fixed average degree equal to a system parameter c. Clusters of the resulting network are treated as nodes at the next hierarchy level and they are connected again at this level and so on, until the process cannot continue. In the RGH model we use all clusters, including those of size 1, when building the next hierarchy level, while in the LRGH model clusters of size 1 stop participating in further steps. We find that in both models the number of nodes at a given hierarchy level h decreases approximately exponentially with h. The height of the hierarchy H, i.e. the number of all hierarchy levels, increases logarithmically with the system size N, i.e. with the number of nodes at the first level. The height H decreases monotonically with the connectivity parameter c in the RGH model and it reaches a maximum for a certain c _max in the LRGH model. The distribution of separate cluster sizes in the LRGH model is a power law with an exponent about ? 1.25. The above results follow from approximate analytical calculations and have been confirmed by numerical simulations.     

Explicit formulae for Chern–Simons invariants of the hyperbolic orbifolds of the knot with Conway’s notation <Emphasis Type="Italic">C</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>, 3)

We calculate the Chern–Simons invariants of the hyperbolic orbifolds of the knot with Conway’s notation C(2n, 3) using the Schläfli formula for the generalized Chern–Simons function on the family of C(2n, 3) cone-manifold structures. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee’s methods. As an application, we calculate the Chern–Simons invariants of cyclic coverings of the hyperbolic C(2n, 3) orbifolds.     

Restricting Positive Energy Representations of Diff<Superscript>+</Superscript>(<Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>) to the Stabilizer of <Emphasis Type="Italic">n</Emphasis> Points

Let G _n ? Diff⁺(S ¹) be the stabilizer of n given points of S ¹. How much information do we lose if we restrict a positive energy representation \(U^c_h\) associated to an admissible pair (c, h) of the central charge and lowest energy, to the subgroup G _n ? The question, and a part of the answer originate in chiral conformal QFT. The value of c can be easily “recovered” from such a restriction; the hard question concerns the value of h. If c ≤ 1, then there is no loss of information, and accordingly, all of these restrictions are irreducible. In this work it is shown that \(U^c_{h}|_{G_n}\) is always irreducible for n =  1 and, if h =  0, it is irreducible at least up to n ≤  3. Moreover, an example is given for c >  2 and certain values of \(h \neq \tilde{h}\) such that \(U^c_{h}|_{G_1}\simeq U^c_{\tilde{h}}|_{G_1}\) . It is also concluded that for these values \(U^c_{h}|_{G_n}\) cannot be irreducible for n ≥  2. For further values of c, h and n, the question is left open. Nevertheless, the example already shows that, on the circle, there are conformal QFT models in which local and global intertwiners are not equivalent.     

Inverse Scattering Problem for a Two Dimensional Random Potential

We study an inverse problem for the two-dimensional random Schrödinger equation (Δ + q + k ²)u = 0. The potential q(x) is assumed to be a Gaussian random function whose covariance operator is a classical pseudodifferential operator. We show that the backscattered field, obtained from a single realization of the random potential q, determines uniquely the principal symbol of the covariance operator of q. The analysis is carried out by combining harmonic and microlocal analysis with stochastic methods.     

Untersuchungen über die Energieabhängigkeit von Kernreaktionen mit Neutronen im Energiebereich zwischen 12 und 19 MeV

Continuing earlier investigations we studied the energy dependence of the cross sections of the following nuclear reactions produced by neutrons in the energy range from 12 to 19 MeV: P³¹(n, 2n)P³², P³¹(n,α)Al²⁸, Cu⁶⁵(n, 2n)Cu⁶⁴, Cu⁶⁵(n, p)Ni⁶⁵, Zn⁶⁴(n, 2n)Zn⁶³, Zn⁶⁴(n, p)Cu⁶⁴, V⁵¹(n, p)Ti⁵¹, Br⁷⁹(n, α)As⁷⁶, O¹⁶(n, α)C¹³. The results were compared with cross sections calculated according to the statistical theory, considering the competition ofγ-ray emission and particle (predominantly neutron) emission from the excited residual nucleus.     

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is

$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left[\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $

for m positive definite operators A _j . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ _p,1 for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ _p,1 for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L ^q (L ^p ) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces – which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

     

Analytic properties of the effective dielectric constant of a two-dimensional Rayleigh model

Analytic properties of the dimensionless static effective dielectric constant f(p, h) of a two-dimensional Rayleigh model (p is the concentration and h is the ratio of the dielectric constants of components) are considered as a function of the complex variable h. It is shown that the only singularities of the function f(p, h) are first-order poles for real h = h _n < 0 (n = 1, 2, ...) with the condensation point h = ?1, which form an infinite discrete (countable) set. The positions of the first ten poles of the function f(p, h) and the residues at these points are calculated and represented graphically versus the concentration. Based on the results obtained, a pole-type approximate formula is proposed that describes the behavior of the function f(p, h) over a wide range of p and complex h.     

Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness b ? a _B are studied in random phase approximation (RPA) and for the less investigated region r _s < 1. Using the single-loop approximation, analytical expressions of the structure factor are obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy in RPA is found to be represented by ? _c (b, r _s ) = α(r _s )/b + β(r _s ) ln(b) + η(r _s ), for small b and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.     

Large Communities in a Scale-Free Network

We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability \(p\in (0,1)\), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph \(G_{t}\) produced by this so-called generalized linear preferential (GLP) model at time t contains a complete subgraph whose vertex set cardinality is given by \(t^\alpha \), where \(\alpha = (1-\varepsilon )\frac{1-p}{2-p}\), for any small \(\varepsilon >0\) asymptotically almost surely.     

Zero-sound branches in asymmetric nuclear matter

A polarization operator constructed in the random phase approximation is used to obtain zero-sound excitations in isospin asymmetric nuclear matter (ANM). Two families of the complex solutions ω_sτ(k),τ= p,n are presented. The imaginary part of the solutions corresponds to the damping of the collective mode due to its overlapping with the particle-hole modes and the subsequent emission of a proton (ω_sp(k)) or a neutron (ω_sn(k)). The dependence of the solutions on the asymmetry parameter is studied.     

Anregungsfunktionen für die (d,p)-, (d,n)-und (d,2n)-Reaktionen an Ce142 bis 11,8 MeV

Total cross sections and excitation functions up to 11·8 MeV have been measured for the Ce¹⁴²(d,p), -(d,n) and -(d, 2n) reactions by the activation method. The cross sections found forE _d=11·8 MeV are 187mb, 54 mb and 535 mb, respectively. By comparing these results with cross sections calculated from the statistical theory of nuclear reactions it can be shown that the (d,p)-reaction and nearly the whole (d,n) -reaction proceed by stripping mechanism.     

Periodic Striped Ground States in Ising Models with Competing Interactions

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value J_c(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of J_c(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.     

Assortativity of complementary graphs

Newman's measure for (dis)assortativity, the linear degree correlationρ _D , is widely studied although analytic insight into the assortativity of an arbitrary network remains far from well understood. In this paper, we derive the general relation (2), (3) and Theorem 1 between the assortativity ρ _D (G) of a graph G and the assortativityρ _D (G ^c) of its complement G ^c. Both ρ _D (G) and ρ _D (G ^c) are linearly related by the degree distribution in G. When the graph G(N,p) possesses a binomial degree distribution as in the Erd?s-Rényi random graphs G _p (N), its complementary graph G _p ^c (N) = G _{1- p} (N) follows a binomial degree distribution as in the Erd?s-Rényi random graphs G _{1- p} (N). We prove that the maximum and minimum assortativity of a class of graphs with a binomial distribution are asymptotically antisymmetric: ρ _max(N,p) = -ρ _min(N,p) for N → ∞. The general relation (3) nicely leads to (a) the relation (10) and (16) between the assortativity range ρ _max(G)–ρ _min(G) of a graph with a given degree distribution and the range ρ _max(G ^c)–ρ _min(G ^c) of its complementary graph and (b) new bounds (6) and (15) of the assortativity. These results together with our numerical experiments in over 30 real-world complex networks illustrate that the assortativity range ρ _max–ρ _min is generally large in sparse networks, which underlines the importance of assortativity as a network characterizer.     

Intertwining Symmetry Algebras of Quantum Superintegrable Systems on Constant Curvature Spaces

A class of quantum superintegrable Hamiltonians defined on a hypersurface in a n+1 dimensional ambient space with signature (p,q) is considered and a set of intertwining operators connecting them are determined. It is shown that the intertwining operators can be chosen such that they generate the su(p,q) and so(2p,2q) Lie algebras and lead to the Hamiltonians through Casimir operators. The physical states corresponding to the discrete spectrum of bound states as well as the degeneration are characterized in terms of some particular unitary representations.     

Photoionization of the Xe atom and Xe@C<Subscript>60</Subscript> molecule

Photoionization of the Xe atom and Xe@C₆₀ molecule have been studied usingthe random phase approximation with exchange (RPAE) method. The Xe atom was described byrelaxed orbitals including overlap integrals. The C₆₀ fullerene has beenrepresented by an attractive short range spherical well with potentialV(r), given byV(r) =  ?V ₀ forr _i  < r < r _o ,otherwise V(r) = 0 wherer _i andr _o are respectively, the inner and outerradii of the spherical shell. The time independent Schrödinger equation was solved usingboth regular and irregular solutions and the continuous boundary conditions atr _i andr _o . The results demonstrate improvementto previous calculations for both the Xe atom and Xe@C₆₀ molecule and comparevery well with the recent experimental data.