Narrow Results

The non-relativistic (scaling) limit of a particle-field Hamiltonian H, called a Dirac–Maxwell operator, in relativistic quantum electrodynamics is considered. It is proven that the non-relativistic limit of H yields a self-adjoint extension of the Pauli–Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics. This is done by establishing in an abstract framework a general limit theorem on a family of self-adjoint operators partially formed out of strongly anticommuting self-adjoint operators and then by applying it to H.

We study a mathematically rigorous derivation of a quantum mechanical Hamiltonian in a general framework. We derive such a Hamiltonian by taking a scaling limit for a generalization of the Nelson model, which is an abstract interaction model between particles and a Bose field with some internal degrees of freedom. Applying it to a model for the field of the nuclear force with isospins, we obtain a Schrödinger Hamiltonian with a matrix-valued potential, the one pion exchange potential, describing an effective interaction between nucleons.

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann–Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (that appeared after the well-known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al. [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.

We study quantum gravitational effect on a two-dimensional open universe with one particle by means of a string bit model. We find that matter is necessarily homogeneously distributed if the influence of the particle on the size of the universe is optimized.

We consider a scaling limit of the Hamiltonian of the generalized spin-boson (GSB) model which is an abstract quantum field theoretical model of particles interacting with a Bose field. Applying it to a Hamiltonian of the field of the nuclear force with isospin, we obtain an effective potential of the interaction between nucleons. Also, we discuss an application to a Hamiltonian of a lattice spin system interacting with a Bose field and obtain a spin–spin interaction in the vacuum of the Bose field. An interaction model between a Fermi field and a Bose field yields an interaction in the vacuum of the Bose field.

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod ${𝕄}_1$ topology instead of Skorohod ${𝕁}_1$ topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the ${𝕁}_1$ topology.

Two types of point particles, large and small, with state space ℝ^d are considered, where d ≥ 2 and "large" and "small" refer to different masses. The small particles move deterministically with very large initial velocities. They transfer their momenta to the large particles through a smooth mean field interaction which completely determines the motion of the large particles. The joint dynamics is described in a spacetime lattice by an Euler scheme for coupled oscillators with a friction term for the large particles. This lattice defines the mesoscale for the system. A scaling limit leads to a replacement of the mesoscale by the macroscale as follows: The very large initial velocities are assumed to be independent and they let a small particle interact with a large particle only for a short time, after which the particle escapes to infinity and new particles start interacting with the large particles. Thus, the initial spatial independence of the small particles causes independence in the time increments of the velocities of the large particles. Therefore, as the friction of the large particles and the speed of the small particles tend to infinity in this scaling limit, the positions of the large particles become the positions of spatially correlated Brownian motions whose correlations can be computed from the interaction force. A similar result holds for a system without friction, where the velocities of the large particles become spatially correlated Brownian motions.

In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all spacetime trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.

Let w (x, t) := (u, v)(x, t), x ∈ ℝ³, t > 0, be the ℝ²-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), u_t(x,0), and v(x,0), v_t(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

We give a new proof of the vanishing noise limit theorem for exit times of one-dimensional diffusions conditioned on exiting through a point separated from the starting point by a potential wall. We also prove a scaling limit for exit location in a model two-dimensional situation.

We consider on the 2D torus the modified Surface Quasi-Geostrophic (mSQG) equation with $L^2$-initial data and perturbed by multiplicative transport noise. Under a suitable scaling of the noises, we show that the solutions converge weakly to the unique solution of the dissipative mSQG equation.

In this paper, we consider the spectrum of a Laplacian matrix, also known as Markov matrices where the entries of the matrix are independent but have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that the variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution converges. We give an expression for the moments of the limiting measure in terms of graph homomorphisms. In some special cases, we identify the limit explicitly. We also study the spectral norm and derive the order of the maximum eigenvalue. We show that our results cover Laplacians of various random graphs including inhomogeneous Erdős–Rényi random graphs, sparse W-random graphs, stochastic block matrices and constrained random graphs.

We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ℝ^d or ℤ^d. The first class consists of random walks on ℤ^d in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2017, with some hints to the main ideas of the proofs. No technical details are presented here.

We present recent work about the scaling limit of random planar maps, which are random graphs embedded in the two-dimensional sphere. We consider a random planar map M_n which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m_n be the vertex set of M_n, which is equipped with the graph distance d_gr. Both when q ≥ 4 is an even integer and when q = 3, there exists a positive constant c_q such that the rescaled metric spaces (m_n; c_qn^−1/4d_gr) converge in distribution in the Gromov-Hausdorff sense, towards a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.