A Note on the Abelian Sandpile in $\pmb{\mathbb{Z}}^{d}$

We analyze the abelian sandpile model on ℤ ^d for the starting configuration of n particles in the origin and 2d−2 particles otherwise. We give a new short proof of the theorem of Fey, Levine and Peres (J. Stat. Phys. 198:143–159, 2010) that the radius of the toppled cluster of this configuration is O(n ^1/d ).     

On the Ornstein-Zernike Behaviour for the Bernoulli Bond Percolation on $\pmb{\mathbb{Z}}^{d},d\geq3$, in the Supercritical Regime

We prove Ornstein-Zernike behaviour in every direction for finite connection functions of bond percolation on ℤ ^d for d≥3 when p, the probability of occupation of a bond, is sufficiently close to 1. Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.     

Non-equilibrium Stationary States in the Symmetric Simple Exclusion with Births and Deaths

We consider the symmetric simple exclusion process in the interval Λ _N :=[−N,N]∩ℤ with births and deaths taking place respectively on suitable boundary intervals I ₊ and I ₋, as introduced in De Masi et al. (J. Stat. Phys. 144:1151–1170, 2011). We study the stationary measure and its macroscopic density profile in the limit N→∞.     

A New Class of Cellular Automata with a Discontinuous Glass Transition

We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density ρ _c for convergence to a completely empty configuration is non trivial, 0<ρ _c <1, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, ρ<ρ _c , emptying always occurs exponentially fast and that ρ _c coincides with the critical density for two-dimensional oriented site percolation on ℤ². This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is discontinuous and at the same time the crossover length diverges faster than any power law. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques. Finally, we will discuss why, due to the peculiar mixed critical/first order character of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D.S. Fisher.     

Extended q -euler numbers and polynomials associated with fermionic p -adic q -integral on Z p

The purpose of this paper is to construct extended q-Euler numbers and polynomials related to fermionic p-adic q-integral on ℤ _p . By evaluating a multivariate p-adic q-integral on ℤ _p , we give new explicit formulas related to these numbers and polynomials.     

Cylindrical cellular automata

A one-dimensional cellular automaton with periodic boundary conditions may be viewed as a lattice of sites on a cylinder evolving according to a local interaction rule. A technique is described for finding analytically the set of attractors for such an automaton. Given any one-dimensional automaton rule, a matrixA is defined such that the number of fixed points on an arbitrary cylinder size is given by the trace ofA ⁿ , where the powern depends linearly on the cylinder size. More generally, the number of strings of arbitrary length that appear in limit cycles of any fixed period is found as the solution of a linear recurrence relation derived from the characteristic equation of an associated matrix. The technique thus makes it possible, for any rule, to compute the number of limit cycles of any period on any cylinder size. To illustrate the technique, closed-form expressions are provided for the complete attractor structure of all two-neighbor rules. The analysis of attractors also identifies shifts as a basic mechanism underlying periodic behavior. Every limit cycle can be equivalently defined as a set of strings on which the action of the rule is a shift of sizes/h; i.e., each string cyclically shifts bys sites inh iterations of the rule. The study of shifts provides detailed information on the structure and number of limit cycles for one-dimensional automata.     

Surface States and Spectra

Let ℤ₊ ^{d +1}= ℤ⁺×ℤ₊, let H ₀ be the discrete Laplacian on the Hilbert space l ²(ℤ₊ ^{d +1}) with a Dirichlet boundary condition, and let V be a potential supported on the boundary ∂ℤ₊ ^{d +1}. We introduce the notions of surface states and surface spectrum of the operator H=H ₀+V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on σ(H ₀) with probability one. To prove this result we combine Aizenman–Molchanov theory with techniques of scattering theory. Received: 18 September 2000 / Accepted: 21 November 2000     

Linear cellular automata and recurring sequences in finite fields

A one-dimensional linear cellular automaton with periodic boundary conditions consists of a lattice of sites on a cylinder evolving according to a linear local interaction rule. Limit cycles for such a system are studied as sets of strings on which the rule acts as a shift of sizes/h; i.e., each string in the limit cycle cyclically shifts bys sites inh iterations of the rule. For any given rule, the size of the shift varies with the cylinder sizen. The analysis of shifts establishes an equivalence between the strings of values appearing in limit cycles for these automata, and linear recurring sequences in finite fields. Specifically, it is shown that a string appears in a limit cycle for a linear automaton rule on a cylinder sizen iff its values satisfy a linear recurrence relation defined by the shift value for thatn. The rich body of results on recurring sequences and finite fields can then be used to obtain detailed information on periodic behavior for these systems. Topics considered here include the inverse problem of identifying the set of linear automata rules for which a given string appears in a limit cycle, and the structure under operations (such as addition and complementation) of sets of limit cycle strings.     

Thomae Formula for General Cyclic Covers of $${{\mathbb {CP}}^1}$$

Let X be a general cyclic cover of \mathbbCP¹{\mathbb{CP}^{1}} ramified at m points, λ₁... λ _m . we define a class of non-positive divisors on X of degree g −1 supported in the pre images of the branch points on X, such that the Riemann theta function does not vanish on their image in J(X). We generalize the results of Bershadsky and Radul (Commun Math Phys 116:689–700, 1988), Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) and Enolskii and Grava (Lett Math Phys 76(2–3):187–214, 2006) and prove that up to a certain determinant of the non-standard periods of X, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point of the curve X. Our approach is based on a refinement of Accola’s results for 3 cyclic sheeted cover (Accola, in Trans Am Math Soc 283:423–449, 1984) and a generalization of Nakayashiki’s approach explained in Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) for general cyclic covers.     

$\mathcal{CPT}$-Symmetric Discrete Square Well

In an addendum to the recent systematic Hermitization of certain N by N matrix Hamiltonians H ^(N)(λ) (Znojil in J. Math. Phys. 50:122105, 2009) we propose an amendment H ^(N)(λ,λ) of the model. The gain is threefold. Firstly, the updated model acquires a natural mathematical meaning of Runge-Kutta approximant to a differential PT\mathcal{PT}-symmetric square well in which P\mathcal{P} is parity. Secondly, the appeal of the model in physics is enhanced since the related operator C\mathcal{C} of the so called “charge” (the requirement of observability of which defines the most popular Bender’s metric Q = PC\Theta=\mathcal{PC}) becomes also obtainable (and is constructed here) in an elementary antidiagonal matrix form at all N. Last but not least, the original phenomenological energy spectrum is not changed so that the domain of its reality (i.e., the interval of admissible couplings λ∈(−1,1)) remains the same.     

A Fredholm Determinant Representation in ASEP

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice ℤ. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers ℤ⁺ and consider the distribution function for the mth particle from the left. In Tracy and Widom (Commun. Math. Phys. 279:815–844, 2008) an infinite series of multiple integrals was derived for the distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.     

Reversibility Problem of Multidimensional Finite Cellular Automata

While the reversibility of multidimensional cellular automata is undecidable and there exists a criterion for determining if a multidimensional linear cellular automaton is reversible, there are only a few results about the reversibility problem of multidimensional linear cellular automata under boundary conditions. This work proposes a criterion for testing the reversibility of a multidimensional linear cellular automaton under null boundary condition and an algorithm for the computation of its reverse, if it exists. The investigation of the dynamical behavior of a multidimensional linear cellular automaton under null boundary condition is equivalent to elucidating the properties of the block Toeplitz matrix. The proposed criterion significantly reduces the computational cost whenever the number of cells or the dimension is large; the discussion can also apply to cellular automata under periodic boundary conditions with a minor modification.     

Sequential Cavity Method for Computing Free Energy and?Surface Pressure

We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice ℤ ^d . Our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of ℤ ^d . Then recent deterministic algorithms for computing marginal probabilities are used to obtain numerical estimates of the quantities of interest. The method works under the assumption of Strong Spatial Mixing (SSP), which is a form of a correlation decay. We illustrate our method on the hard-core and monomer-dimer models, on which we improve several earlier estimates. For example we show that the exponential of the monomer-dimer coverings of ℤ³ belongs to the interval [0.78595,0.78599], improving best previously known estimate of [0.7850,0.7862] obtained in (Friedland and Peled in Adv. Appl. Math. 34:486–522, 2005; Friedland et al. in J. Stat. Phys., 2009). Moreover, we show that given a target additive error ε>0, the computational effort of our method for these two models is (1/ε) ^O(1) both for the free energy and surface pressure values. In contrast, prior methods, such as the transfer matrix method, require exp ((1/ε) ^O(1)) computation effort.     

On SLE Martingales in Boundary WZW Models

Following Bettelheim et al. (Phys Rev Lett 95:251601, 2005), we consider the boundary WZW model on a half-plane with a cut growing according to the Schramm–Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions for boundary correlation functions to be SLE martingales. Necessary conditions come from the requirement for the boundary field at the tip of the cut to have a depth two null vector. Sufficient conditions are established using Knizhnik–Zamolodchikov equations for boundary correlators. Combining these two approaches, we show that in the case of G = SU(2) the boundary correlator is an SLE martingale if and only if the boundary field carries spin 1/2. In the case of G = SU(n) and the level k = 1, there are several situations when boundary one-point correlators are SLE _κ -martingales. If the boundary field is labelled by the defining n-dimensional representation of SU(n), we obtain \varkappa = 2{\varkappa=2} . For n even, by choosing the boundary field labelled by the (unique) self-adjoint fundamental representation, we get \varkappa = 8/(n + 2){\varkappa=8/(n {+} 2)} . We also study the situation when the distance between the two boundary fields is finite, and we show that in this case the SLE_\varkappa{{\rm SLE}_\varkappa} evolution is replaced by SLE_\varkappa,r{{\rm SLE}_{\varkappa,\rho}} with r = \varkappa -6{\rho=\varkappa -6} .     

Non-uniqueness Results for Critical Metrics of Regularized Determinants in Four Dimensions

The regularized determinant of the Paneitz operator arises in quantum gravity [see Connes in (Noncommutative geometry, 1994), IV.4.γ]. An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of conformally covariant differential operators, q-curvature and holography. Progress in Mathematics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge laplacian on p-forms (p < n/2). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997).     

Wegner-type Bounds for a Two-particle Lattice Model with a Generic “Rough” Quasi-periodic Potential

In this paper, we consider a class of two-particle tight-binding Hamiltonians, describing pairs of interacting quantum particles on the lattice ℤ ^d , d ≥ 1, subject to a common external potential V(x) which we assume quasi-periodic and depending on auxiliary parameters. Such parametric families of ergodic deterministic potentials (“grands ensembles”) have been introduced earlier in Chulaevsky (2007), in the framework of single-particle lattice systems, where it was proved that a non-uniform analog of the Wegner bound holds true for a class of quasi-periodic grands ensembles. Using the approach proposed in Chulaevsky and Suhov (Commun Math Phys 283(2):479–489, 2008), we establish volume-dependent Wegner-type bounds for a class of quasi-periodic two-particle lattice systems with a non-random short-range interaction.     

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let λ _d (p) be the p monomer-dimer entropy on the d-dimensional integer lattice ℤ ^d , where p∈[0,1] is the dimer density. We give upper and lower bounds for λ _d (p) in terms of expressions involving λ _d−1(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of ℤ ^d is bounded above by λ _d (p). We compute the first three terms in the formal asymptotic expansion of λ _d (p) in powers of  \frac1d\frac{1}{d}. We prove that the lower asymptotic matching conjecture is satisfied for λ _d (p). Converted to a power series in p, our “formal” expansion shows remarkable validity in low dimensions, d=1,2,3, in which dimensions we give some numerical studies.     

The CPT Group of the Spin-3/2 Field

We find out that both the matrix and the operator CPT groups for the spin-3/2 field (with or without mass) are respectively isomorphic to D ₄⋊ℤ₂ and Q×ℤ₂. These groups are exactly the same groups as for the Dirac field, though there is no a priori reason why they should coincide.     

Quantum Brownian Motion on Non-Commutative Manifolds: Construction,Deformation and Exit Times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A_q{\mathcal{A}_\theta} , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_q{\mathcal{A}_\theta} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.