Narrow Results

In this paper, we investigate both well-posedness and existence of an invariant measures for a class of wave equations defined on higher-dimensional integer set ${\mathbb{Z}}^N$ driven by a family of infinite-dimensional superlinear. By a space-time cut-off technique we prove the global existence as well as uniqueness of solutions in $(L^2(\mathrm{\Omega },{ℱ},{\ell }^2)\cap L^p(\mathrm{\Omega },{ℱ},{\ell }^p))\times L^2(\mathrm{\Omega },{ℱ},{\ell }^2)$ with $p\ge 2$ for a family of nonlinear drift terms of polynomial growth of arbitrary order as well as a family of nonlinear diffusion terms with superlinear growth rate. In addition, the existence of invariant measures is established in ${\ell }^2\times {\ell }^2$ by formulating Krylov–Bogolyubov’s method. The tightness of a family of distribution laws of solutions is derived via the idea of uniform tail-estimates of solutions in order to overcome the lack of compactness on infinite lattices.

We give an intuitive method — using local, cyclic replica symmetry — to isolate exponential tree decay in truncated (connected) correlations. We give an expansion and use the symmetry to show that all terms vanish, except those displaying replica condensation. The condensation property ensures exponential tree decay.

We illustrate our method in a low-temperature Ising system, but expect that one can use a similar method in other random field and quantum field problems. While considering the illustration, we prove an elementary upper bound on the entropy of random lattice surfaces.

We review the derivation of the Gaudin model with integrable boundaries. Starting from the non-symmetric R-matrix of the inhomogeneous spin-½ XXZ chain and generic solutions of the reflection equation and the dual reflection equation, the corresponding Gaudin Hamiltonians with boundary terms are calculated. An alternative derivation based on the so-called classical reflection equation is discussed.

We study the capillary condensation of a lattice gas with nearest neighbor attraction confined to a slit-like pore filled with frozen obstacles (matrix). First, the lattice Monte Carlo simulations were performed for a slit-like pore without obstacles. Next, the pore filled with obstacles, i.e., confined microporous medium, is prepared by adsorbing and then by quenching a hard sphere fluid in the pore of the width H. The model includes fluid-wall attraction; however, the fluid-matrix interaction is entirely repulsive. We have investigated how concentration of obstacles and the pore width H influence the capillary phase diagrams. Our numerical experiments reveal essential changes in the critical temperature and in the shape of the coexistence line with the amount of quenched obstacles.

Some observable consequences that follow from a fiber bundle description of a tight binding condensed matter system on a lattice are described. The geometrical picture can be extended to describe non-periodic lattice structures, where a single Brillouin zone is predicted.