Mathematical Physics, Analysis and Geometry最新文献

Using Riemann–Hilbert methods, we establish a Tracy–Widom like formula for the generating function of the occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order three. We also obtain a non linear coupled heat equation. Combining these two equations we obtain a PDE for the logarithm of the the generating function of the Pearcey process.

In this paper, we study the HC-model with a countable set (mathbb Z) of spin values on a Cayley tree of order (kge 2). This model is defined by a countable set of parameters (that is, the activity function (lambda _i>0), (iin mathbb Z)). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:

• Let (Lambda =sum _ilambda _i). For (Lambda =+infty ) there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);

• For (Lambda <+infty ), the uniqueness of TIGM is proved;

• Let (Lambda _textrm{cr}(k)=frac{k^k}{(k-1)^{k+1}}). If (0<Lambda le Lambda _textrm{cr}), then there is exactly one TPGM that is TIGM;

• For (Lambda >Lambda _textrm{cr}), there are exactly three TPGMs, one of which is TIGM.

We show that Rieffel’s quantum Gromov–Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov–Hausdorff distance applied to the associated state spaces.

In this paper we consider the Blume–Emery–Griffiths model on Cayley trees. We reduce the problem of describing the splitting Gibbs measures of the Blume–Emery–Griffiths model to the description of the solutions of some algebraic equation. Also, we analyse the set of translation-invariant splitting Gibbs measures for a two parametric BEG model on Cayley trees.

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice (mathbb {Z}^d) in dimensions (d>4), in the vicinity of the critical point, and prove an upper bound (|x|^{-(d-2)}exp [-c|x|/xi ]), where the correlation length (xi ) has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions (d{>}4) has a “plateau.” We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.

We consider a four-prototype Rossler system introduced by Otto Rössler among others as prototypes of the simplest autonomous differential equations (in the sense of minimal dimension, minimal number of parameters, minimal number of nonlinear terms) having chaotic behavior. We contribute towards the understanding of its chaotic behavior by studying its integrability from different points of view. We show that it is neither Darboux integrable, nor (C^1)-integrable.

In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the d-dimensional body-centered cubic (BCC) lattice ({mathbb {L}^d}) and the set of non-negative integers ({{mathbb {Z}}_+}). Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on ({mathbb {L}^d} times {{mathbb {Z}}_+}) in all dimensions (dge 9). As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value (-1).

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions

where (|q_{pm }|=1) and (q_{+}=delta q_{-}), (sigma delta =-1). In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region (-6<xi <6) with (xi =frac{x}{t}). In this paper, we give the asymptotic expansion of the solution q(x, t) for other solitonic regions (xi <-6) and (xi >6). Based on the Riemann–Hilbert formulation of the Cauchy problem, further using the ({bar{partial }}) steepest descent method, we derive different long time asymptotic expansions of the solution q(x, t) in above two different space-time solitonic regions. In the region (xi <-6), phase function (theta (z)) has four stationary phase points on the ({mathbb {R}}). Correspondingly, q(x, t) can be characterized with an ({mathcal {N}}(Lambda ))-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function (textrm{Im}nu (zeta _i)). In the region (xi >6), phase function (theta (z)) has four stationary phase points on (i{mathbb {R}}), the corresponding asymptotic approximations can be characterized with an ({mathcal {N}}(Lambda ))-soliton with diverse residual error order ({mathcal {O}}(t^{-1})).

We study extensions of the classical Toda lattices at several different space–time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant–Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel–Lie algebras. Our study brings into play factorizations of Loewner–Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space–time scales. Along the way we derive a novel, minimal box–ball system for the Full Kostant–Toda Lattice that does not involve any capacities or colorings, and which has a natural interpretation in terms of the Robinson–Schensted–Knuth algorithm. We provide as well an extension of O’Connell’s ordinary differential equations to the Full Kostant–Toda Lattice.

In some recent literature the role of non self-adjoint Hamiltonians, (Hne H^dagger ), is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.