Volume 20, Issue 1

Abstract. It is shown that gauge field-dependent fermion Dirac operators from lattice QCD form an ergodic operator family in the probabilistic sense, provided the gauge field is an ergodic random field. As a consequence, the integrated density of states of such Dirac operators in the thermodynamic limit exists and is almost surely independent of the chosen gauge field configuration.

Abstract. We consider the Wigner ensemble of nn$n\times n$ real symmetric random matrices A(n)$A^{(n)}$ whose entries are determined by independent identically distributed random variables aij,ij${\lbrace a_{ij}, i\le j\rbrace }$ that have symmetric probability distribution with variance v2$v^2$ and study the asymptotic behavior of the spectral norm A(n)$\Vert A^{(n)}\Vert $ as n$n\rightarrow \infty $. We prove that if the moment 𝐄aij12+20${\bf E}\,\vert a_{ij}\vert ^{12+2\delta _0}$ with any strictly positive 0$\delta _0$ exists, then the probability 𝐏A(n)>2v(1+xn-2/3)${{\bf P}\lbrace \Vert A^{(n)} \Vert > 2v(1+xn^{-2/3}) \rbrace }$, x>0${x>0}$, is bounded in the limit of infinite n$n$ by an expression that does not depend on the details of the probability distribution of aij$a_{ij}$. The proof is based on the completed and modified version of the approach developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.

Abstract. Regularity of solutions is studied for backward stochastic parabolic Ito equations. An analog of the second fundamental inequality (second energy estimate) and the related existence theorem are obtained for domains with boundary. This result leads to a representation theorem for non-Markov processes in bounded domains and other applications.