Results 1  10
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35
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 65 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Ground state properties of the Nelson Hamiltonian  A Gibbs measurebased approach
 Rev. Math. Phys
, 2001
"... The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in positi ..."
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Cited by 19 (12 self)
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The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.
Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions
, 2008
"... We construct Euclidean random fields X over IR d, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Sc ..."
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Cited by 17 (12 self)
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We construct Euclidean random fields X over IR d, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo–differential operators (− ∆ + m 2 0) −α for α ∈ (0,1) and m0> 0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X = Gα ∗ F, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property.
INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND POINCARÉ INEQUALITIES
"... Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1. ..."
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Cited by 14 (6 self)
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Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1.
Functional Integral Representation of the PauliFierz Model with Spin 1/2
, 706
"... A FeynmanKactype formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e−tHPF generated by the PauliFierz Hamiltonian with spin 1/2 in nonrelativistic quantum electrody ..."
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Cited by 6 (6 self)
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A FeynmanKactype formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e−tHPF generated by the PauliFierz Hamiltonian with spin 1/2 in nonrelativistic quantum electrodynamics is constructed. When no external potential is applied HPF turns translation invariant and it is decomposed as a direct integral HPF = ∫ ⊕ R3 HPF(P)dP. The functional integral representation of e−tHPF(P) is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived. 1 2 The PauliFierz model with spin 1
Convergence in total variation on Wiener chaos
 STOCH. PROC. APPL
"... Let {Fn} be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F ∞ satisfying Var(F∞)> 0. Our first result is a sequential version of a theorem by Shigekawa [25]. More precisely, we prove, without additional assump ..."
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Cited by 5 (1 self)
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Let {Fn} be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F ∞ satisfying Var(F∞)> 0. Our first result is a sequential version of a theorem by Shigekawa [25]. More precisely, we prove, without additional assumptions, that the sequence {Fn} actually converges in total variation and that the law of F ∞ is absolutely continuous. We give an application to discrete nonGaussian chaoses. In a second part, we assume that each Fn has more specifically the form of a multiple WienerItô integral (of a xed order) and that it converges in L2 (Ω) towards F∞. We then give an upper bound for the distance in total variation between the laws of Fn and F∞. As such, we recover an inequality due to Davydov and Martynova [6]; our rate is weaker compared to [6] (by a power of 1/2), but the advantage is that our proof is not only sketched as in [6]. Finally, in a third part we show that the convergence in the celebrated PeccatiTudor theorem actually holds in the total variation topology.
A modified Poincaré inequality and its application to first passage percolation
, 2006
"... We extend a functional inequality for the Gaussian measure on R n to the one on R N. This inequality improves on the classical Poincaré inequality for Gaussian measures. As an application, we prove that First Passage Percolation has sublinear variance when the edge times distribution belongs to a w ..."
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Cited by 4 (2 self)
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We extend a functional inequality for the Gaussian measure on R n to the one on R N. This inequality improves on the classical Poincaré inequality for Gaussian measures. As an application, we prove that First Passage Percolation has sublinear variance when the edge times distribution belongs to a wide class of continuous distributions, including the exponential one. This extends a result by Benjamini, Kalai and Schramm [3], valid for positive Bernoulli edge times.
Lectures on Gaussian approximations with Malliavin calculus
, 2013
"... Overview. In a seminal paper of 2005, Nualart and Peccati [40] discovered a surprising central limit theorem (called the “Fourth Moment Theorem ” in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is ..."
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Cited by 4 (3 self)
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Overview. In a seminal paper of 2005, Nualart and Peccati [40] discovered a surprising central limit theorem (called the “Fourth Moment Theorem ” in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor [46] gave a multidimensional version of this characterization. Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and OrtizLatorre [39], giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper [27] (written in collaboration with Peccati) in which, by bringing together Stein’s method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the Fourth Moment Theorem. It turns out that Stein’s method and Malliavin calculus fit together admirably well. Their interaction has led to some remarkable new results involving central and noncentral limit theorems for functionals of infinitedimensional Gaussian fields. The current survey aims to introduce the main features of this recent theory. It originates from
CHAOS, CONCENTRATION, AND MULTIPLE VALLEYS
, 2008
"... Disordered systems are an important class of models in statistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include various models of glasses and polymers. They also arise in other areas, like fitness models in evolu ..."
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Cited by 3 (0 self)
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Disordered systems are an important class of models in statistical mechanics, having the defining characteristic that the energy landscape is a fixed realization of a random field. Examples include various models of glasses and polymers. They also arise in other areas, like fitness models in evolutionary biology. The ground state of a disordered system is the state with minimum energy. The system is said to be chaotic if a small perturbation of the energy landscape causes a drastic shift of the ground state. We present a rigorous theory of chaos in disordered systems that confirms longstanding physics intuition about connections between chaos, anomalous fluctuations of the ground state energy, and the existence of multiple valleys in the energy landscape. Combining these results with mathematical tools like hypercontractivity, we establish the existence of the above phenomena in eigenvectors of GUE matrices, the KauffmanLevin model of evolutionary biology, directed polymers in random environment, a subclass of the generalized