large-N Behavior Research Articles

Phase structure of the quartic-cubic generalized two dimensional Yang-Mills U(N) on the sphere

The large-N behavior of the quartic-cubic generalized two dimensional Yang-Mills U(N) on the sphere is investigated for small cubic couplings. It is shown that single transition at the critical area, which is present for the quartic model, is split into two transitions; both of them are third order. The phase diagram of the system for small cubic couplings is obtained.

Metastability in interacting nonlinear stochastic differential equations: II. Large-N behaviour

We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N2), the system synchronizes, in the sense that all particles assume almost the same position in their respective local potential most of the time. In a previous work (Berglund et al 2007 Nonlinearity 20 2551), we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system's stationary configurations. We analysed, for arbitrary N, the behaviour for coupling intensities slightly below the synchronization threshold. Here we describe the behaviour for any positive coupling intensity γ of order N2, provided the particle number N is sufficiently large (as a function of γ/N2). In particular, we determine the transition time between synchronized states, as well as the shape of the ‘critical droplet’, to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded.

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Consider a large system of N Brownian motions in \({\mathbb{R}}^d\) with some non-degenerate initial measure on some fixed time interval [0,β] with symmetrised initial-terminal condition. That is, for any i, the terminal location of the i-th motion is affixed to the initial point of the σ(i)-th motion, where σ is a uniformly distributed random permutation of 1,...,N. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature 1/β. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the N paths) and of the mean of the normalised occupation measures of the N motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel–Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker–Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of \({\rm e}^{-\beta {\mathcal{H}}_N}\) , where \({\mathcal{H}}_N\) is an N-particle Hamilton operator in a trap.

New Monte Carlo method for planar Poisson–Voronoi cells

By a new Monte Carlo algorithm, we evaluate the sidedness probability pn of a planar Poisson–Voronoi cell in the range 3 ⩽ n ⩽ 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behaviour of pn as n → ∞. Our pn values all have between four and six significant digits. Accurate n dependent averages, second moments and variances are obtained for the cell area and the cell perimeter. The numerical large-n behaviour of these quantities is analysed in terms of an asymptotic power series in n−1. Snapshots are shown of typical occurrences of extremely rare events, implicating cells of up to n = 1600 sides embedded in an ordinary Poisson–Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.

Large deviations for trapped interacting Brownian particles and paths

We introduce two probabilistic models for N interacting Brownian motions moving in a trap in ℝd under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the path-repellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose–Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross–Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross–Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.

Phase transitions of large-N two-dimensional Yang–Mills and generalized Yang–Mills theories in the double scaling limit

The large-N behavior of Yang–Mills and generalized Yang–Mills theories in the double-scaling limit is investigated. By the double-scaling limit, it is meant that the area of the manifold on which the theory is defined, is itself a function of N. It is shown that phase transitions of different orders occur, depending on the functional dependence of the area on N. The finite-size scalings of the system are also investigated. Specifically, the dependence of the dominant representation on A, for large but finite N is determined.

Perturbative versus nonperturbative dynamics of the fuzzy S2× S2

We study a matrix model with a cubic term, which incorporates both the fuzzy S^2*S^2 and the fuzzy S^2 as classical solutions. Both of the solutions decay into the vacuum of the pure Yang-Mills model (even in the large-N limit) when the coefficient of the cubic term is smaller than a critical value, but the large-N behavior of the critical point is different for the two solutions. The results above the critical point are nicely reproduced by the all order calculations in perturbation theory. By comparing the free energy, we find that the true vacuum is given either by the fuzzy S^2 or by the ``pure Yang-Mills vacuum'' depending on the coupling constant. In Monte Carlo simulation we do observe a decay of the fuzzy S^2*S^2 into the fuzzy S^2 at moderate N, but the decay probability seems to be suppressed at large N. The above results, together with our previous results for the fuzzy CP^2, reveal certain universality in the large-N dynamics of four-dimensional fuzzy manifolds realized in a matrix model with a cubic term.

Self-similar variational perturbation theory for critical exponents

We extend field theoretic variational perturbation theory by self-similar approximation theory, which greatly accelerates convergence. This is illustrated by recalculating the critical exponents of O (N) -symmetric phi(4) theory. From only three-loop perturbation expansions in 4-epsilon dimensions, we obtain analytic results for the exponents, which are close to those derived recently from ordinary field-theoretic variational perturbational theory to seventh order. In particular, the specific-heat exponent is found to be in good agreement with best-measured exponent alpha approximately -0.0127 of the specific-heat peak in superfluid helium, found in a satellite experiment. In addition, our analytic expressions reproduce also the exactly known large- N behavior of the exponents.

N-ality and topology at finite temperature

We study [JHEP 0309 (2003) 034] the spectrum of confining strings in SU(3) pure gauge theory, in different representations of the gauge group. Our results provide direct evidence that the string spectrum agrees with predictions based on n-ality. We also investigate [L. Del Debbio, H. Panagopoulos, E. Vicari, hep-th/0407068, to appear in JHEP] the large-N behavior of the topological susceptibility χ in four-dimensional SU(N) gauge theories at finite temperature, and in particular across the finite-temperature transition at Tc. The results indicate that χ has a nonvanishing large-N limit for T < Tc, as at T = 0, and that the topological properties remain substantially unchanged in the low-temperature phase. On the other hand, above the deconfinement phase transition, χ shows a large suppression. The comparison between the data for N = 4 and N = 6 hints at a vanishing large-N limit for T > Tc.

Topological susceptibility of SU(N) gauge theories at finite temperature

We investigate the large-N behavior of the topological susceptibility in four-dimensional SU(N) gauge theories at finite temperature, and in particular across the finite-temperature transition at Tc. For this purpose, we consider the lattice formulation of the SU(N) gauge theories and perform Monte Carlo simulations for N=4,6. The results indicate that the topological susceptibility has a nonvanishing large-N limit for T<Tc, as at T=0, and that the topological properties remain substantially unchanged in the low-temperature phase. On the other hand, above the deconfinement phase transition, the topological susceptibility shows a large suppression. The comparison between the data for N=4 and N=6 hints at a vanishing large-N limit for T>Tc.

Systematic Finite-Sampling Inaccuracy in Free Energy Differences and Other Nonlinear Quantities

Systematic inaccuracy is inherent in any computational estimate of a non-linear average, such as the free energy difference ΔF between two states or systems, because of the availability of only a finite number of data values, N. In previous work, we outlined the fundamental statistical description of this “finite-sampling error.” We now give a more complete presentation of (i) rigorous general bounds on the free energy and other nonlinear averages, which generalize Jensen's inequality; (ii) asymptotic N→∞ expansions of the average behavior of the finite-sampling error in ΔF estimates; (iii) illustrative examples of large-N behavior, both in free-energy and other calculations; and (iv) the universal, large-N relation between the average finite-sampling error and the fluctuation in the error. An explicit role is played by Levy and Gaussian limiting distributions.

EXACT FORMULA FOR THE MEAN LENGTH OF A RANDOM WALK ON THE SIERPINSKI TOWER

We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula [Formula: see text] The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by [Formula: see text] This also establishes the leading large-n behavior [Formula: see text] that may be expected on general grounds, where Nn is the number of sites on the nth generation tower and [Formula: see text] is the spectral dimension of the fractal.

Ultraslow vacancy-mediated tracer diffusion in two dimensions: the Einstein relation verified.

We study the dynamics of a charged tracer particle (TP) on a two-dimensional lattice, all sites of which except one (a vacancy) are filled with identical neutral, hard-core particles. The particles move randomly by exchanging their positions with the vacancy, subject to the hard-core exclusion. In the case when the charged TP experiences a bias due to external electric field E (which favors its jumps in the preferential direction), we determine exactly the limiting probability distribution of the TP position in terms of appropriate scaling variables and the leading large-n (n being the discrete time) behavior of the TP mean displacement X(n); the latter is shown to obey an anomalous, logarithmic law /X(n)/=alpha(0)(/E/)ln(n). Comparing our results with earlier predictions by Brummelhuis and Hilhorst [J. Stat. Phys. 53, 249 (1988)] for the TP diffusivity D(n) in the unbiased case, we infer that the Einstein relation mu(n)=betaD(n) between the TP diffusivity and the mobility mu(n)=lim(/E/-->0)(/X(n)///E/n) holds in the leading n order, despite the fact that both D(n) and mu(n) are not constant but vanish as n--> infinity. We also generalize our approach to the situation with very small but finite vacancy concentration rho(v), in which case we find a ballistic-type law /X(n)/=pi(alpha)(0)(/E/)rho(v)n. We demonstrate that here, again, both D(n) and mu(n), calculated in the linear in rho(v) approximation, do obey the Einstein relation.

Closed string tachyons, AdS/CFT, and large N QCD

We find that tachyonic orbifold examples of AdS/CFT have corresponding instabilities at small radius, and can decay to more generic gauge theories. We do this by computing a destabilizing Coleman-Weinberg effective potential for twisted operators of the corresponding quiver gauge theories, generalizing calculations of Tseytlin and Zarembo and interpreting them in terms of the large-N behavior of twisted-sector modes. The dynamically generated potential involves double-trace operators, which affect large-N correlators involving twisted fields but not those involving only untwisted fields, in line with large-N inheritance arguments. We point out a simple reason that no such small radius instability exists in gauge theories arising from freely acting orbifolds, which are tachyon-free at large radius. When an instability is present, twisted gauge theory operators with the quantum numbers of the large-radius tachyons aquire VEVs, leaving a gauge theory with fewer degrees of freedom in the infrared, analogous to but less extreme than ``decays to nothing'' studied in other systems with broken supersymmetry. In some cases one is left with pure glue QCD plus decoupled matter and U(1) factors in the IR, which we thus conjecture is described by the corresponding (possibly strongly coupled) endpoint of tachyon condensation in the M/String-theory dual.

Large-n critical behavior of O(n)×O(m) spin models

We consider the Landau-Ginzburg-Wilson Hamiltonian with O(n)x O(m) symmetry and compute the critical exponents at all fixed points to O(n^{-2}) and to O(\epsilon^3) in a \epsilon=4-d expansion. We also consider the corresponding non-linear sigma model and determine the fixed points and the critical exponents to O(\tilde{\epsilon}^2) in the \tilde{\epsilon}=d-2 expansion. Using these results, we draw quite general conclusions on the fixed-point structure of models with O(n)xO(m) symmetry for n large and all 2 < d < 4.

D0-branes as light-front confined quarks

We argue that different aspects of Light-Front QCD at confined phase can be recovered by the Matrix Quantum Mechanics of D0-branes. The concerning Matrix Quantum Mechanics is obtained from dimensional reduction of pure Yang-Mills theory to 0+1 dimension. The aspects of QCD dynamics which are studied in correspondence with D0-branes are: 1) phenomenological inter-quark potentials, 2) whiteness of hadrons and 3) scattering amplitudes. In addition, some other issues such as the large-N behavior, the gravity--gauge theory relation and also a possible justification for involving ``non-commutative coordinates'' in a study of QCD bound-states are discussed.

Generalized conformal symmetry and oblique AdS-CFT correspondence for matrix theory

The large-N behaviour of matrix theory is discussed on the basis of the previously proposed generalized conformal symmetry. The concept of `oblique' AdS-CFT correspondence, in which the conformal symmetry involves both the spacetime coordinates and the string coupling constant, is proposed. Based on the explicit predictions for two-point correlators, possible implications for the matrix-theory conjecture are discussed.

Rigorous Estimates of the Tails of the Probability Distribution Function for the Random Linear Shear Model

In previous work Majda and McLaughlin, and Majda computed explicit expressions for the 2Nth moments of a passive scalar advected by a linear shear flow in the form of an integral over R N . In this paper we first compute the asymptotics of these moments for large moment number. We are able to use this information about the large-N behavior of the moments, along with some basic facts about entire functions of finite order, to compute the asymptotics of the tails of the probability distribution function. We find that the probability distribution has Gaussian tails when the energy is concentrated in the largest scales. As the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer, and the distribution moves smoothly from Gaussian through exponential and “stretched exponential.” We also show that the derivatives of the scalar are increasingly intermittent, in agreement with experimental observations, and relate the exponents of the scalar derivative to the exponents of the scalar.

The Maximum of a Random Walk and Its Application to Rectangle Packing

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate,where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Quasi-Bound States of Two Magnons in the Spin-[ 1/2]XXZ Chain

We study two-magnon Bethe states in the spin-1/2 XXZ chain. The string hypothesis assumes that complex rapidities of the bound states take special forms. It is known, however, that there exist “non-string states,” which substantially disagrees with the string hypothesis. In order to clarify their nature, we study the large-N behavior of solutions of the Bethe-Ansatz equations to obtain explicit forms of typical Bethe states, where N is the length of the chain, and apply the scaling analysis (the multifractal analysis) to the Bethe states. It turns out that the non-string states contain “quasi-bound” states, which in some sense continuously interpolate between extended states and localized states. The “quasi-bound” states can be distinguished from known three types of states, i.e., extended, localized, and critical states. Our results indicate that there might be a need to reconsider the standard classification scheme of wavefunctions.