On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph

We prove that AB site percolation occurs on the line graph of the square lattice when $p \in (1 - \sqrt {1 - p_c } ,\sqrt {1 - p_c } )$ , where p _c is the critical probability for site percolation in $\mathbb{Z}^2$ . Also, we prove that AB bond percolation does not occur on $\mathbb{Z}^2$ for p = $\frac{1}{2}$ .     

Rationality of Bershadsky-Polyakov Vertex Algebras

We prove the conjecture of Kac-Wakimoto on the rationality of exceptional W-algebras for the first non-trivial series, namely, for the Bershadsky-Polyakov vertex algebras ${W_3^{(2)}}$ W 3 ( 2 ) at level k =  p/2?3 with ${p = 3, 5, 7, 9, \dots}$ p = 3 , 5 , 7 , 9 , ? . This gives new examples of rational conformal field theories.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

Let ${\cal A}$ be the affine space of all connections in an SU(2) principal fibre bundle over ?³. The set of homogeneous isotropic connections forms a line l in ${\cal A}$ . We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding $l \hookrightarrow {\cal A}$ does not continuously extend to an embedding $\overline{l} \hookrightarrow \overline{\cal A}$ of the respective compactifications. Here, the Bohr compactification $\overline{l}$ corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and $\overline{\cal A}$ to that of loop quantum gravity. Analogous results are given for the anisotropic case.     

A second look at a controversial percolation exponent—Is η negative in three dimensions?

Series for the three dimensional bond percolation probability are reexamined in the light of a recent evaluation of the percolation thresholdp _c . We find $$\beta = 0.435 \pm 0.035$$ which implies $$\eta = - 0.01 \pm 0.09$$ We determine Δ₁ = 1.05 ± 0.15 and also observe a second correction exponent near Δ₂ ~ 1.7.     

Statistics of directed self-avoiding walks on percolation clusters at the percolation threshold

We here study directed self-avoiding walks on site diluted square lattice at the percolation threshold by two parameter real space renormalization group method. We found \(v_\parallel ^{p_c } = 1.00\) and \(v_ \bot ^{p_c } = 0.4348\) from cell-to-cell transformation method. This \(v_ \bot ^{p_c } \) value is then compared with the modified Alexander-Orbach formula that \(v_ \bot ^{p_c } = {{d_S } \mathord{\left/ {\vphantom {{d_S } {2d_L }}} \right. \kern-0em} {2d_L }}\) whered _s is the fracton dimension andd _L is the spreading dimension of the infinite directed percolation cluster.     

The Percolation Transition in the Zero-Temperature Domany Model

We analyze a deterministic cellular automaton σ ^?=(σ ⁿ :n≥0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice $\mathbb{N}$ . The state space $\mathcal{S}_\mathbb{H} = \left\{ { - 1, + 1} \right\}^\mathbb{H}$ consists of assignments of ?1 or +1 to each site of $\mathbb{H}$ and the initial state $\sigma ^0 = \left\{ {\sigma _{^x }^0 } \right\}_{x \in \mathbb{H}}$ is chosen randomly with P(σ ⁰ _x=+1)=p∈[0,1]. The sites of $\mathbb{H}$ are partitioned in two sets $\mathcal{A}$ and $\mathcal{B}$ so that all the neighbors of a site x in $\mathcal{A}$ belong to $\mathcal{B}$ and vice versa, and the discrete time dynamics is such that the σ ^? _x 's with ${x \in \mathcal{A}}$ (respectively, $\mathcal{B}$ ) are updated simultaneously at odd (resp., even) times, making σ ^? _x agree with the majority of its three neighbors. In ref. 1 it was proved that there is a percolation transition at p=1/2 in the percolation models defined by σ ⁿ , for all times n∈[1,∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν, and η of the dependent percolation models defined by σ ⁿ , n∈[1,∞], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice).     

Percolation with Constant Freezing

We introduce and study a model of percolation with constant freezing (PCF) where edges open at constant rate $1$ , and clusters freeze at rate $\alpha $ independently of their size. Our main result is that the infinite volume process can be constructed on any amenable vertex transitive graph. This is in sharp contrast to models of percolation with freezing previously introduced, where the limit is known not to exist. Our interest is in the study of the percolative properties of the final configuration as a function of $\alpha $ . We also obtain more precise results in the case of trees. Surprisingly the algebraic exponent for the cluster size depends on the degree, suggesting that there is no lower critical dimension for the model. Moreover, even for $\alpha <\alpha _c$ , it is shown that finite clusters have algebraic tail decay, which is a signature of self organised criticality. Partial results are obtained on $\mathbb {Z}^d$ , and many open questions are discussed.     

Influence of assortativity and degree-preserving rewiring on the spectra of networks

Newman’s measure for (dis)assortativity, the linear degree correlation coefficient $\rho _{D}$ , is reformulated in terms of the total number N _k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases $\rho _{D}$ conform our desired objective. Spectral metrics (eigenvalues of graph-related matrices), especially, the largest eigenvalue $\lambda _{1}$ of the adjacency matrix and the algebraic connectivity $\mu _{N-1}$ (second-smallest eigenvalue of the Laplacian) are powerful characterizers of dynamic processes on networks such as virus spreading and synchronization processes. We present various lower bounds for the largest eigenvalue $\lambda _{1}$ of the adjacency matrix and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that the lower bounds for $\lambda _{1}$ increase with $\rho _{D}$ . A new upper bound for the algebraic connectivity $\mu _{N-1}$ decreases with $\rho _{D}$ . Applying the degree-preserving rewiring algorithm to various real-world networks illustrates that (a) assortative degree-preserving rewiring increases $\lambda _{1}$ , but decreases $\mu _{N-1}$ , even leading to disconnectivity of the networks in many disjoint clusters and that (b) disassortative degree-preserving rewiring decreases $\lambda _{1}$ , but increases the algebraic connectivity, at least in the initial rewirings.     

Gaussian limit for critical oriented percolation in high dimensions

In this paper, we consider the spread-out oriented bond percolation models inZ ^d ×Z withd>4 and the nearest-neighbor oriented bond percolation model in sufficiently high dimensions. Let η _n ,n=1, 2, ..., be the random measures defined onR ^d by $$\eta _n (A) = \sum\limits_{x \in Z^d } {1_A (x/\sqrt n )1_{\{ (0,0) \to (x,n)\} } } $$ The mean of η _n , denoted by $\bar \eta _n $ , is the measure defined by $$\bar \eta _n (A) = E_p [\eta _n (A)]$$ We use the lace expansion method to show that the sequence of probability measures $[\bar \eta _n (R^d )]^{ - 1} \bar \eta _n $ converges weakly to a Gaussian limit asn→∞ for everyp in the subcritical regime as well as the critical regime of these percolation models. Also we show that for these models the parallel correlation length $\xi (p)~|p_c - p|^{ - 1} $ asp?p_c     

Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions

We extend the method of Balister, Bollobás and Walters (Phys. Rev. E 76:011110, 2007) for determining rigorous confidence intervals for the critical threshold of two dimensional lattices to three (and higher) dimensional lattices. We describe a method for determining a full confidence interval and apply it to show that the critical threshold for bond percolation on the simple cubic lattice is between \(0.2485\) and \(0.2490\) with \(99.9999\,\%\) confidence, and the critical threshold for site percolation on the same lattice is between \(0.3110\) and \(0.3118\) with \(99.9999\,\%\) confidence.     

Just Renormalizable TGFT’s on U(1) d with Gauge Invariance

We study the polynomial Abelian or U(1) ^d Tensorial Group Field Theories equipped with a gauge invariance condition in any dimension d. We prove the just renormalizability at all orders of perturbation of the ${\varphi^4_6}$ and ${\varphi^6_5}$ random tensor models. We also deduce that the ${\varphi^4_5}$ tensor model is super-renormalizable.     

Rate of Convergence for Cardy’s Formula

We show that crossing probabilities in 2D critical site percolation on the triangular lattice in a piecewise analytic Jordan domain converge with power law rate in the mesh size to their limit given by the Cardy–Smirnov formula. We use this result to obtain new upper and lower bounds of ${e^{O(\sqrt{{\rm log}\,{\rm log} R})}\, R^{-1/3}}$ for the probability that the cluster at the origin in the half-plane has diameter R, improving the previously known estimate of R ^?1/3+o(1).     

On Energy-Momentum Transfer of Quantum Fields

We prove the following theorem on bounded operators in quantum field theory: if \({\|[B,B^*(x)]\|\leqslant{\rm const}D(x)}\) , then \({\|B^k_\pm(\nu)G(P^0)\|^2\leqslant{\rm const}\int D(x - y){\rm d}|\nu|(x){\rm d}|\nu|(y)}\) , where D(x) is a function weakly decaying in spacelike directions, \({B^k_\pm}\) are creation/annihilation parts of an appropriate time derivative of B, G is any positive, bounded, non-increasing function in \({L^2(\mathbb{R})}\) , and \({\nu}\) is any finite complex Borel measure; creation/annihilation operators may be also replaced by \({B^k_t}\) with \({\check{B^k_t}(p)=|p|^k\check{B}(p)}\) . We also use the notion of energy-momentum scaling degree of B with respect to a submanifold (Steinmann-type, but in momentum space, and applied to the norm of an operator). These two tools are applied to the analysis of singularities of \({\check{B}(p)G(P^0)}\) . We prove, among others, the following statement (modulo some more specific assumptions): outside p = 0 the only allowed contributions to this functional which are concentrated on a submanifold (including the trivial one—a single point) are Dirac measures on hypersurfaces (if the decay of D is not to slow).     

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points \(N(J)\) in a region \(J\) in \(\mathbb R^d\) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of \(N(J)\) tends to infinity as \(|J| \rightarrow \infty \) . This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for \(\{E(k;J)\}\) —the probabilities of there being exactly \(k\) points in \(J\) —all lie on the negative real \(z\) -axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the \(k\) -th largest eigenvalue at the soft edge, and of the spacing between \(k\) -th neighbors in the bulk.     

Uniform Existence of the Integrated Density of States for Randomly Weighted Hamiltonians on Long-Range Percolation Graphs

We consider random Hamiltonians defined on long-range percolation graphs over $\mathbb {Z}^{d}$ . The Hamiltonian consists of a randomly weighted Laplacian plus a random potential. We prove uniform existence of the integrated density of states and express the IDS using a Pastur-Shubin trace formula.     

Uniqueness and Nondegeneracy of Positive Solutions of {(-\Delta)^s u + u = u^p \, {\rm in} \, \mathbb{R}^N} when s is Close to 1

We consider the equation ${(-\Delta)^s u + u = u^p}$ , with ${s \in (0, 1)}$ in the subcritical range of p. We prove that if s is sufficiently close to 1 the equation possesses a unique minimizer, which is nondegenerate.     

Strong magnetic fields,Dirichlet boundaries,and spectral gaps

We consider magnetic Schrödinger operators $$H(\lambda \vec a) = ( - i\nabla - \lambda \vec a(x))^2$$ inL ₂(R ⁿ ), where $\vec a \in C^1 (R^n ;R^n )$ and λεR. LettingM={x;B(x)=0}, whereB is the magnetic field associated with $\vec a$ , and $M_{\vec a} = \{ x;\vec a(x) = 0\}$ , we prove that $H(\lambda \vec a)$ converges to the (Dirichlet) Laplacian on the closed setM in the strong resolvent sense, as λ→∞,provided the set $M\backslash M_{\vec a}$ has measure zero. In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that $M\backslash M_{\vec a}$ has measure zero). As a consequence, if we are given a periodic fieldB where the regions withB=0 have non-empty interior and are enclosed by the region withB≠0, magnetic wells will be created when λ is large, opening up gaps in the spectrum of $H(\lambda \vec a)$ . We finally address the question of absolute continuity of $\vec a$ for periodic $H(\vec a)$ .