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Prediction theory and fourier series in several variables. II

The concept of freeness was introduced by Voiculescu in the context of operator algebras. Later it was observed that it is also relevant for large random matrices. We will show how the combination of various free probability results with a linearization trick allows to address successfully the problem of determining the asymptotic eigenvalue distribution of general selfadjoint polynomials in independent random matrices.

Free Probability and Random Matrices

LetX be a finite connectedCW-complex. Suppose that its fundamental group π is residually finite, i.e. there is a nested sequence ... ⊂ Г
 m
 + 1 ⊂ Г
 m
 ⊂ ... ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL
2-Betti number ofX is the limit of the sequenceb
p(Xm)/[π:Г
 m
 ] whereb
p(Xm) is the (ordinary)p-th Betti number of the finite covering ofX associated with Г
 m
 .

ApproximatingL 2-invariants by their finite-dimensional analogues

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular graphs. The general answer involves a quantity for infinite graphs that we call ‘tree entropy’, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle [Ann. Probab. 21 1329–1371].

Asymptotic Enumeration of Spanning Trees

Determinants, First Minors, and Cofactors.- A Summary of Basic Determinant Theory.- Intermediate Determinant Theory.- Particular Determinants.- Further Determinant Theory.- Applications of Determinants in Mathematical Physics.

http://inis.jinr.ru/sl/m_mathematics/ma_algebra/mal_linear%20algebra/vein%20r.,%20dale%20p.%20determinants%20and%20their%20applications%20in%20mathematical%20physics%20(springer,%201998)(isbn%200387985581)(393s).pdf

Determinants and their applications in mathematical physics

Murray and J. v. Neumann classified factors (i.e., central rings of operators) by means of a "relative dimension function." They studied, extensively, those factors for which this dimension function has a finite range ("finite factors") and showed that these factors (and these factors alone) admit a trace function' with the standard algebraic properties. In an attempt to establish, what seems to us to be a further important algebraic property of the trace, viz., the trace of a generalized nilpotent operator is zero, and, more generally, the trace of an operator lies in the convex hull of its spectrum, we were led to the introduction of a determinant theory for finite factors. This paper will be concerned, principally, with the development of this theory. We might note that it is a simple algebraic matter to prove that the trace T(N) of a proper nilpotent N is zero. In fact, if N' = 0 and E is the projection on the closure of the range of N, then EN = N so that (NE)'-' = N'-'E = 0. Then T(N) = T(EN) = T(NE) = 0, by induction on n. That the normalized trace lies in the convex hull of the spectrum of a finite-dimensional matrix follows at once by bringing the matrix to super-diagonal form, whereby the normalized trace appears as the "center of gravity" of the spectrum. This fact together with the theory developed in R.O. IV, Chapter IV, yields the same result for operators in an approximately finite factor. Furthermore, it is an immediate consequence of the spectral theorem that the trace of a normal operator in an arbitrary finite factor lies in the convex hull of its spectrum. None of these easily proved facts enabled us to conclude the result for arbitrary non-normal operators in nonapproximately finite factors. However, the general result was established as a byproduct of the determinant theory. In ?2 we define the determinant on regular operators in a factor of type II1,, and establish the properties of this determinant. The proof that the trace lies in the convex hull of the spectrum is given in ?3 as an application of the results of ?2. The uniqueness of the determinant is established in ?4 by means of an algebraic characterization. The final section, ?5, begins with a discussion of the normalization which has taken place in the definition of the determinant. A

https://www.jstor.org/stable/pdfplus/1969645.pdf

Bent Fuglede

Papers

Determinant theory in finite factors