The Scaling Limit of Lattice Trees in High Dimensions

We prove that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion known as integrated super-Brownian excursion (ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof uses the lace expansion. Received: 27 September 1996 / Accepted: 28 July 1997     

Thermodynamic Limit for a Spin Lattice

An integrable spin lattice is a higher dimensional generalization of integrable spin chains. In this paper we consider a special spin lattice related to quantum mechanical interpretation of the three-dimensional lattice model in statistical mechanics (Zamolodchikov and Baxter). The integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables. The significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters. There is a specific bilinear functional equation for the eigenvalues of this transfer matrix.The spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity. The limiting form of bilinear equation is derived. It allows to analyze the distributions of eigenvalues of the whole commuting set. The ground state distribution is obtained explicitly. A structure of excited states is discussed.     

Fractal formations in the Lattice Limit Cycle model

We examine the fractal patterns arising in the Lattice Limit Cycle model, when it is restricted on square and fractal lattices. We show that, for processes taking place on regular 2d substrates, the fractal dimensions depend on the kinetic constants and we have observed a sharp phase-transition from uniform 2d spatial distributions (d_f=2) when the kinetic parameters are near the Hopf bifurcation point, to a inside the limit cycle region. For processes taking place on substrates which contain inactive sites, we observe nucleation of homologous species around inactive regions leading to poisoning, when the active sites are distributed in a fractal manner on the substrate. This is less frequent in cases where the active sites are distributed uniformly and randomly on the lattice leading, normally, to non-trivial steady states.     

Large Deviations and Scaling Limit

We explore the behavior under scaling limits of large systems using methods from the theory large deviations. This is carried out through the examination of a few examples.     

On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice ${h\mathbb{Z}}$ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}$ with the fractional Laplacian (?Δ) ^α as dispersive symbol. In particular, we obtain that fractional powers ${\frac{1}{2} < \alpha < 1}$ arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian ?Δ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.     

Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit

In this paper, we construct an adiabatic invariant for a large 1–d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.     

Derivation of the Leroux System as the Hydrodynamic Limit of a Two-Component Lattice Gas

The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, even beyond the appearance of shocks. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following [4, 5], the stochastic version of Tartar–Murat theory of compensated compactness is extended to two-component stochastic models.Supported in part by the Hungarian Science Foundation (OTKA), grants T26176 and T037685.     

Lattice models with N=2 supersymmetry

We introduce lattice models with explicit N=2 supersymmetry. In these interacting models, the supersymmetry generators Q+/- yield the Hamiltonian H=(Q(+),Q(-)) on any graph. The degrees of freedom can be described as either fermions with hard cores, or as quantum dimers; the Hamiltonian of our simplest model contains a hopping term and a repulsive potential. We analyze these models using conformal field theory, the Bethe ansatz, and cohomology. The simplest model provides a manifestly supersymmetric lattice regulator for the supersymmetric point of the massless (1+1)-dimensional Thirring (Luttinger) model. Generalizations include a quantum monomer-dimer model on a two-leg ladder.     

The Hierarchy Principle and the Large Mass Limit of the Linear Sigma Model

In perturbation theory we study the matching in four dimensions between the linear sigma model in the large mass limit and the renormalized nonlinear sigma model in the recently proposed flat connection formalism. We consider both the chiral limit and the strong coupling limit of the linear sigma model. Our formalism extends to Green functions with an arbitrary number of pion legs, at one loop level, on the basis of the hierarchy as an efficient unifying principle that governs both limits. While the chiral limit is straightforward, the matching in the strong coupling limit requires careful use of the normalization conditions of the linear theory, in order to exploit the functional equation and the complete set of local solutions of its linearized form.     

Diffusion on the Scaling Limit of the Critical Percolation Cluster in the Diamond Hierarchical Lattice

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on-diagonal heat kernel for the percolation clusters and for the underlying lattice. In this setting a number of features of the lattice are inherited by the critical cluster.     

Large N chiral dynamics

Some properties of large N chiral dynamics are discussed, using an effective Lagrangian that has been derived by Rosenzweig, Schechter, and Trahern; Di Vecchia and Veneziano; and Nath and Arnowitt.     

Large Deviations in Weakly Interacting Boundary Driven Lattice Gases

One-dimensional, boundary-driven lattice gases with local interactions are studied in the weakly interacting limit. The density profiles and the correlation functions are calculated to first order in the interaction strength for zero-range and short-range processes differing only in the specifics of the detailed-balance dynamics. Furthermore, the effective free-energy (large-deviation function) and the integrated current distribution are also found to this order. From the former, we find that the boundary drive generates long-range correlations only for the short-range dynamics while the latter provides support to an additivity principle recently proposed by Bodineau and Derrida.     

Large Deviations in Quantum Lattice Systems: One-Phase Region

We give large deviation upper bounds, and discuss lower bounds, for the Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the lattice. We cover general interactions and general observables, both in the high temperature regime and in dimension one.     

Large Detuning Limit for the Multipartite Systems Interacting with Electromagnetic Fields

We study the dynamics of the multipartite systems nonresonantly interacting with electromagnetic fields, focusing on the large detuning limit for the effective Hamiltonian. Due to the many-particle interference effects, the more rigorous large detuning condition for neglecting the rapidly oscillating terms for the effective Hamiltonian should be N 1/2 g, instead of g usually used in the literature even in the case of multipartite systems, with N the number of microparticles involved, g the coupling strength, the detuning. This result is significant since merely the satisfaction of the original condition will result in the invalidity of the effective Hamiltonian and the errors of the parameters associated with the detuning in the multipartite case.     

倒格子不同角度的引入

介绍了从几个不同的方面引入倒格子的数学方法及有关步骤.     

Pion Distribution Amplitude from the Lattice

Results for the second moment of the pion distribution amplitude in quenched QCD are reported, and compared with other QCD-based predictions as well as experimental determinations. A broad agreement is found, which calls for a more precise lattice computation.     

Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions (called “watermelons” with a wall) on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyze the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for $L\ll O(\sqrt{N})$ and $L\geq O(\sqrt{N})$ . A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value $L=L_{c}(N)\sim\sqrt{N}$ . This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L～L _c (N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L?L _c (N)] and the left [L?L _c (N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L～L _c (N) and atypical fluctuations in the right tail L?L _c (N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices. This paper is an extended version of (Schehr et al. in Phys. Rev. Lett. 101:150601, 2008) and (Forrester et al. in Nucl. Phys. B 844:500–526, 2011).     

Thermodynamical and Chiral Limit in Computer Simulations of Lattice Quant um Field Theories

The finite-size effects which appear in computer simulations of chiral-symmetry breaking in lattice gauge theory are analyzed. To reduce these systematic errors, a better method for extrapolating the chiral condensate to the infinite volume and chiral limit is proposed.     

Large Deviations of Lattice Hamiltonian Dynamics Coupled to Stochastic Thermostats

We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary states as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling.