On the meaning of Parisi's functional order parameter

The author has recently proved that a famous formula discovered by G. Parisi gives at any temperature the correct value for the limiting free energy of a large class of mean field models for spin glasses (a class which contains in particular the Sherrington–Kirkpatrick model). Here we prove rigorously that (generically) the “functional order parameter” occuring in this formula can be interpreted as predicted by Parisi, namely as representing the limiting distribution of the overlap of two independent configurations. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Cauchy transforms of self-similar measures: the Laurent coefficients

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in . Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {a_n}_n=0^∞ of F. We give asymptotic formulas for {a_n}_n=0^∞ and for F^(k)(z) near the support of μ, hence the precise growth rates on |a_n| and |F^(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.     

Convexity of trace functionals and Schrödinger operators

Let H be a semi-bounded self-adjoint operator on a separable Hilbert space. For a certain class of positive, continuous, decreasing, and convex functions F we show the convexity of trace functionals of the form tr(F(H+U−ε(U)))−ε(U), where U is a bounded, self-adjoint operator and ε(U) is a normalizing real function—the Fermi level—which may be identical zero. If additionally F is continuously differentiable, then the corresponding trace functional is Fréchet differentiable and there is an expression of its gradient in terms of the derivative of F. The proof of the differentiability of the trace functional is based upon Birman and Solomyak's theory of double Stieltjes operator integrals. If, in particular, H is a Schrödinger-type operator and U a real-valued function, then the gradient of the trace functional is the quantum mechanical expression of the particle density with respect to an equilibrium distribution function f=−F^′. Thus, the monotonicity of the particle density in its dependence on the potential U of Schrödinger's operator—which has been understood since the late 1980s—follows as a special case.     

Maximization,under equality constraints,of a functional of a probability distribution

Let $\text{F}$ be a family of probability distributions. Let O, C₁…C_n be real functions on $\text{F}$. Let z₁…z_n be real numbers. Then we consider the problem of maximization of the object function O(F)(F?$\text{F}$) under the equality constraints C₁(F)=z₁(i=1,…,n) . The theory is developed in order to solve problems of the following kind: Find the maximal variance of a stop-loss reinsured risk under partial information on the risk such as its range and two first moments.     

The inverse balayage problem for Markov chains

Let X be a Markov chain, let A be a finite sunset of its countable state space. let ?A consist of states in A′ that can be reached in one step from A and let v be a prescribed probability measure on ?A. In this paper we study the following inverse exit problem: describe and analyze the set M(v) of probability measures μ on A such that P^μ {X(T)?·}=v(·) where T= inl{k: X(k)?A′} is the first exit time from A. Characterizations are provided for elements of M(v), extreme points of M(v) and those measures in M(v) that are maximal with respect to a partial ordering induced by excessive functions. Potential theoretic aspects of the problem and one-dimensional birth and death processes are treated in detail, and examples are given that illustrate implications and limitations of the theory.     

Bernstein theorems and transformations of correlation measures in statistical physics

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products (·, ?) and the maps K, K ^?1 which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): F(γ) → F?(γ) = {F(α?β)}α,β?γ. For the operators F?(γ) and F?(γ), we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function F(γ).     

Hamilton-Jacobi semi-groups in infinite dimensional spaces

Let (X,ρ) be a Polish space endowed with a probability measure μ. Assume that we can do Malliavin Calculus on (X,μ). Let be a pseudo-distance. Consider QtF(x)=infy∈X{F(y)+d²(x,y)/2t}. We shall prove that QtF satisfies the Hamilton-Jacobi inequality under suitable conditions. This result will be applied to establish transportation cost inequalities on path groups and loop groups in the spirit of Bobkov, Gentil and Ledoux.     

Moments of Two-Variable Functions and the Uniqueness of Graph Limits

For a symmetric bounded measurable function W on [0, 1]² and a simple graph F, the homomorphism density $t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$ can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (G _n ) of dense graphs is said to be convergent if the probability, t(F, G _n ), that a random map from V(F) into V(G _n ) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]². Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

Bound on integrals: Elimination of the dual and reduction of the number of equality constraints

Let L_j (j = 1, …, n + 1) be real linear functions on the convex set $\text{F}$ of probability distributions. We consider the problem of maximization of L_n+1(F) under the constraint F ? $\text{F}$ and the equality constraints L₁(F) = z₁ (i = 1, …, n). Incorporating some of the equality constraints into the basic set $\text{F}$, the problem is equivalent to a problem with less equality constraints. We also show how the dual problems can be eliminated from the statement of the main theorems and we give a new illuminating proof of the existence of particular solutions.The linearity of the functions L_j(j = 1, …, n + 1) can be dropped in several results.     

Studying discrete dynamical systems through differential equations

In this paper we consider dynamical systems generated by a diffeomorphism F defined on U an open subset of Rn, and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation, , also defined on U. In particular the case where F has n−1 functionally independent first integrals is considered. In this case X is constructed by imposing that it shares with F the same set of first integrals and that the functional equation μ(F(x))=det(DF(x))μ(x), x∈U, has some non-zero solution, μ. Several examples for n=2,3 are presented, most of them coming from several well-known difference equations.     

Variational measures and the Kurzweil-Henstock integral

For a given continuous function F on a compact interval E in the set ? of reals the problem is how to describe the “total change” of F on a set M ? E. Full variational measures W _F (M) and V _F (M) (see Section 2) in the sense presented by B. S. Thomson are introduced in this work to this aim. They are generated by two slightly different interval functions, namely the oscillation of F over an interval and the value of the additive interval function generated by F, respectively. They coincide with the concept of classical total variation if M is an interval and they are zero if on the set M the function F is of negligible variation. The Kurzweil-Henstock integration is shortly described and some of its properties are studied using the variational measure W _F (M) for the indefinite integral F of an integrable function f.     

Rank procedures for some two-population multivariate extended classification problems

Given independent samples from three multivariate populations with cumulative distribution functions F⁽¹⁾(x), F⁽²⁾(x), and F⁽⁰⁾(x) = θF⁽¹⁾(x) + (1 ? θ)F⁽²⁾(x), where 0 ≤ θ ≤ 1 is unknown, the three-action problem involving decision as to whether the value of θ is high, low, or intermediate, is considered. A class of consistent procedures based on the relative spacing of three sample averages of linearly compounded rank scores is formulated. The asymptotic operating characteristics of the procedures when F⁽¹⁾ and F⁽²⁾ come close together are studied and the best choice of the compounding coefficients in terms of these considered. The consequence of using estimates of the best coefficients on the asymptotic operating characteristics is also examined.     

Une série génératrice des formes paraboliques de Maass

A certain periodic function F_μ,ν(z), an eigenfunction of the Laplacian on the upper half-plane with respect to z depending on some parameters μ, ν, is not automorphic, but the function μ → F_μ,ν(z)−F_μ,ν(−1/0 extends to a larger domain than the function F_μ,ν(z) itself. Consequently, at the poles of this latter function of μ, the coefficients of the polar parts provide non-analytic modular forms: all Maass cusp forms are finite linear combinations of forms obtained in this way, allowing the second parameter ν to vary     

Probability and Fourier Duality for Affine Iterated Function Systems

Let d be a positive integer, and let μ be a finite measure on ? ^d . In this paper we ask when it is possible to find a subset Λ in ? ^d such that the corresponding complex exponential functions e _λ indexed by Λ are orthogonal and total in L ²(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L ²(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ? ^d ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.     

On the small risk approximation

A small risk is a random variable whose deviation from its mean is small in some sense, so that we may replace, for example, X by μ+z(X − μ) and expand related functions and random variables in terms of powers of z. Several models are revisited (the problem of Pareto optimal risk exchanges, equilibrium, the reinsurer's monopoly and the Bowley solution) and approximate solutions are found. The last model (chains of reinsurance) is new.     

Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R⁺. This family forms an orthogonal basis for the subspace of L-invariant functions in L²(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L²(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R⁺.     

Right Markov Processes and Systems of Semilinear Equations with Measure Data

In the paper we prove the existence of probabilistic solutions to systems of the form ?Au = F(x, u) + μ, where F satisfies a generalized sign condition and μ is a smooth measure. As for A we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on L¹. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer’s property (L) of Markov processes and in terms of order compactness of the associated resolvent.     

Fuzzy sets as a basis for a theory of possibility

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U = {u} which is characterized by its membership function μ_F, then a proposition of the form $\text{“X}\text{is}\text{F”}$, where X is a variable taking values in U, induces a possibility distribution $\text{t}\text{?}{}_{\mathrm{x}}$ which equates the possibility of X taking the value u to μ_F(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution $\text{t}\text{?}{}_{\mathrm{x}}$ in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle.A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form $\text{“X}\text{is}\text{F}$ is α-possible”, where α is a number in the interval [0,1], is formulated and illustrated by examples.     

Functions, Measures, and Equipartitioning Convex k-Fans

A k-fan in the plane is a point x∈?² and k halflines starting from x. There are k angular sectors σ ₁,…,σ _k between consecutive halflines. The k-fan is convex if every sector is convex. A (nice) probability measure μ is equipartitioned by the k-fan if μ(σ _i )=1/k for every sector. One of our results: Given a nice probability measure μ and a continuous function f defined on sectors, there is a convex 5-fan equipartitioning μ with f(σ ₁)=f(σ ₂)=f(σ ₃).