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Traces of Semigroups Associated with Interacting Particle Systems Sergio

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Traces of Semigroups Associated with Interacting Particle Systems Sergio Traces of Semigroups Associated with Interacting Particle Systems Sergio Albeverio, Alexei Daletskii, Alexander Kalyuzhnyi no. 264 Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs- gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer- sität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2006 Traces of semigroups associated with interacting particle systems Sergio Albeverio Inst. Ang. Math., Universit¨atBonn; BiBoS; SFB 611; IZKS; CERFIM (Locarno); Acc. Arch. (Mendrisio) Alexei Daletskii College of Science and Technology The Nottingham Trent University U.K. Alexander Kalyuzhnyi Institute of Mathematics Kiev, Ukraine January 3, 2006 Abstract Witten Laplacians associated with interacting particle systems on infinite coverings of compact manifolds are considered. The probablilistic represen- tations of the corresponding heat kernels are given. Von Neumann traces of the corresponding semigroups are computed. 2000 AMS Mathematics Subject Classification. Primary: 60G55, 58A10, 58A12. Secondary: 58B99 Key words: Witten Laplacian, infinite covering, von Neumann algebra, Betti numbers, Configuration space, Gibbs measure Contents 1 Introduction 1 2 Configuration spaces and measures 4 3 Random Witten Laplacian 5 4 Probabilistic representations of the heat kernels 8 5 Von Neumann algebras associated with configuration spaces 15 6 Appendix: Gibbs measures on configuration spaces. 21 1 Introduction The relationship between traces of semigroups associated with elliptic opera- tors acting in spaces of differential forms on a compact Riemannian manifold X, and geometrical and topological properties of these manifolds, is a deep and important fact of global analysis. In the case of Laplace type opera- tors, the supertrace of the corresponding heat semigroup does not depend on time and is equal to the index of the corresponding Dirac operator (the McKean-Singer identity). On the other hand, its short time asymptotics can be expressed in terms of geometric quantities of X, which leads to the Gauss-Bonnet-Chern theorem, see e.g. [13] and [18]. Different versions of this relationship are investigated in the framework of the Atiyah-Singer In- dex Theorem and its more recent generalizations, see e.g. [9]. In the case where X is not compact but admits an infinite group G of isometries with compact quotient, the usual operator trace is substituted by the trace TrA in the commutant A of the action of G (which is a von Neu- mann algebra). This leads to the notion of L2-invariants of X (in particular, M. Atiyah’s L2-Betti numbers and the Novikov-Shubin invariants). The cor- responding theory for perturbed Laplacians (Witten Laplacians associated with measures on X) leads to important results such as (asymptotic) Morse inequalities. In the case where the perturbation is G-invariant, the corre- 1 sponding semigroup Tt belongs to A, and its trace is given by the formula Z TrA Tt = tr K(x; x; t)dx; (1) X=G where K(t; x; y) is the integral kernel of Tt and tr is the usual matrix trace. The right-hand side of the latter formula is called sometimes the Θ-function associated with H, and can be interpreted as the Laplace transform of the spectral density function of H. For a review on L2-invariants and related questions see [28], [29]. On the other hand, there is an extensive theory of random Schr¨odinger operators H = ∆ + q, i.e. Laplace operators with potentials q given by homogeneous random fields (see [30], [21]). That is, q is supposed to be G-invariant in the sense that q(gx; !) = q(x; Ug!); (2) where G 3 g 7! Ug is a representation of G by measure preserving trans- formations of the probability space. For such operators, the Θ-function is defined by the formula Z Θ(t) = E tr K(t; x; x)dx; (3) X=G and can be interpreted in the following way. Let HΛ be a self-adjoint re- striction of H to a compact domain Λ. Then, under certain conditions on ¡tHΛ 1 ¡tHΛ the potential, e is a trace-class operator, limΛ"X volΛ Tr e is a non- random quantity, and 1 Θ(t) = lim Tr e¡tHΛ : (4) Λ"X volΛ Here Tr is the usual operator trace. Thus, heuristically, Θ(t) coincides with a regularized trace of the semigroup e¡tH . Sometimes it can be interpreted as a trace in a certain von Neumann algebra (for instance if the potential is a periodic or an almost periodic function). The aim of the present paper is to consider the case where H is the Witten Lapalcian associated with a random measure σ on a Riemannian manifold X, and the probability space is the space ΓX of locally finite configurations γ in X equipped with a Gibbs measure ¹. 2 The measure σ has the form ¡Eγ (x) σγ(dx) = e dx; (5) P where Eγ(x) = y2γ v(½(x; y)), ½ is the distance on X and v is a smooth function with compact support. Measures of such type appear, via the gener- alized Mecke identity, in the theory of configuration spaces, and in particular in the theory of Laplace operators on differential forms over ΓX (see [4], [5], [3]). In fact, the Witten Laplacian H associated with σ is a ”part” of the Hodge-de Rham operator on ΓX associated with the Gibbs measure ¹. The structure of the latter operator is very complicated in the case where ¹ is different from the Poisson measure. We believe that the study of spectral properties of H, which is a more realistic goal, then the study of the full Hodge-de Rham operator on ΓX , may already give interesting links between the properties of ¹ and geometrical and topological properties of X. Let us remark that the interest in the analysis on infinite configuration spaces has risen in recent years, because of new approaches and rich appli- cations in statistical mechanics and quantum field theory (see [6], [7] and the review [31]). L2-Betti numbers of configuration spaces with Poisson and Lebesgue-Poisson measures, were computed in [2] and [15] respectively (see also [1], [14]). The operator H is related to the following model from statistical mechan- ics. Let us consider a particle with position x performing a random motion in X and interacting with a random media (gas) described by the Gibbs measure ¹. The distribution of the particle is given by the random measure σγ(dx); where Eγ(x) is the energy of the interaction of the particle x and the configuration γ of gas particles. Let us remark that spectral density- and theta-functions of certain Schr¨odingeroperators related to particles in a random media have been discussed in [30] (see also references therein). Our situation of Witten operators acting in spaces of differential forms is significantly more complicated because of the specific structure of the corre- sponding potential, which is a matrix-valued function, neither positive nor bounded. As above, we suppose that there exists an infinite group G of isometries of X such that X=G is a compact Riemannian manifold. The main result of the present paper is the proof of an upper bound for the corresponding theta-function. In particular, we show that Θ(t) is 3 finite for all times t. Moreover, we interpret it as a trace in a certain von Neumann algebra, and show that the corresponding analog of the McKean- Singer formula holds. Acknowledgments. We are very grateful to A. Eberle, K. D. Elworthy, Yu. Kondratiev, X.-M. Li, E. Lytvinov, S. Paycha, M. R¨ockner for interest- ing discussions and useful remarks. One of the authors (AD) would like to thank the Department of Stochastics , University of Bonn, for the hospitality. Financial support by SFB 611 is gratefully acknowledged. 2 Configuration spaces and measures Let X be a complete connected, oriented, C1 Riemannian manifold of infi- nite volume with a lower bounded curvature. We assume that there exists an infinite discrete group of isometries of X such that X=G is a compact Riemannian manifold. The configuration space ΓX over X is defined as the set of all locally finite subsets (configurations) in X: ΓX := fγ ½ X j jγ \ Λj < 1 for each compact Λ ½ Xg : (6) Here, jAj denotes the cardinality of a set A. We can identify any γ 2 ΓX with the positive, integer-valued Radon measure X "x 2 M(X); (7) x2γ P where "x is the Dirac measure with mass at x, x2? "x :=zero measure, and M(X) denotes the set of all positive Radon measures on the Borel σ-algebra of X. The space ΓX is endowed with the relative topology as a subset of the space M(X) with the vague topology, i.e., the weakest topology on ΓX with respect to which all maps Z X ΓX 3 γ 7! hf; γi := f(x) γ(dx) ´ f(x) (8) X x2γ are continuous. Here, f 2 C0(X) (:=the set of all continuous functions on X with compact support). 4 The action of G on X can be lifted to a diagonal action on ΓX : ΓX 3 γ = f:::; x; y; z; :::g 7! gγ = f:::; gx; gy; gz; :::g 2 ΓX ; (9) g 2 G. Let ¹ be a Gibbs measure on ΓX (see Appendix). We assume that: (i) ¹ satisfies the Ruelle bound, that is, ¯ ¯ ¯ (n)¯ n k¹ · a (10) (n) for some constant a, where k¹ is the n-th correlation function of ¹; (ii) ¹ is invariant w.r.t. to the G-action (9). A class of Gibbs measures which satisfy these properties is described in Appendix (see Remark 9). 3 Random Witten Laplacian 2 1 Let v 2 C0 (R ) with supp v ½ [¡r; r] ; where r > 0 is the injectivity radius of X, and define the function V : X £ X ! R1 by V (x; y) = v(½(x; y)); (11) x; y 2 X, where ½ is the Riemannian distance on X.
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