Berezin integrals and Poisson processes

Angelis, Gian Fabrizio De; Jona-Lasinio, G.; Sidoravičius, Vladas

doi:10.1088/0305-4470/31/1/026

Cited by 20 publications

(42 citation statements)

References 11 publications

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“…In this case the convergence of perturbation theory plays a major role on the way to rigorous results. Recently, it has been possible to give a true probabilistic expression to general Grassmaniann integrals in terms of discrete jump processes (Poisson processes) [40], [41] so that classical probability may become a main tool also in the study of fermionic systems especially in view of developing non perturbative methods. For an early example of connection between anticommutative calculus and probability see [42].…”

Section: Discussionmentioning

confidence: 99%

Renormalization group and probability theory

Jona-Lasinio

2001

Physics Reports

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The renormalization group has played an important role in the physics of the second half of the twentieth century both as a conceptual and a calculational tool. In particular it provided the key ideas for the construction of a qualitative and quantitative theory of the critical point in phase transitions and started a new era in statistical mechanics. Probability theory lies at the foundation of this branch of physics and the renormalization group has an interesting probabilistic interpretation as it was recognized in the middle seventies. This paper intends to provide a concise introduction to this aspect of the theory of phase transitions which clarifies the deep statistical significance of critical universality.

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Section: Discussionmentioning

confidence: 99%

Renormalization group and probability theory

Jona-Lasinio

2001

Physics Reports

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“…In this Section, we specialize the discussion to the uniformly fully connected models defined by Eqs. (1)(2)(3). For these models, both the sets T and L have a single element, namely T = (M − 1)η/ǫ and λ = 1, whereas we may count, in general, M distinct values V in the set V .…”

Section: Uniformly Fully Connected Modelsmentioning

confidence: 99%

The exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

Ostilli

Presilla

2006

J. Stat. Mech.

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Abstract. By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condensates into the subspace spanned by the states of the system associated with the lowest potential level. Exact ground state for a class of matrix Hamiltonian models 2

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“…The perturbative parameters p (2) are determined by the system of equations (54) which, according to (73) and (77), reads…”

Section: Equations For the Perturbative Parameters: Second Ordermentioning

confidence: 99%

A perturbative probabilistic approach to quantum many-body systems

2013

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Abstract. In the probabilistic approach to quantum many-body systems, the ground-state energy is the solution of a nonlinear scalar equation written either as a cumulant expansion or as an expectation with respect to a probability distribution of the potential and hopping (amplitude and phase) values recorded during an infinitely lengthy evolution. We introduce a perturbative expansion of this probability distribution which conserves, at any order, a multinomial-like structure, typical of uncorrelated systems, but includes, order by order, the statistical correlations provided by the cumulant expansion. The proposed perturbative scheme is successfully tested in the case of pseudo-spin 1/2 hard-core boson Hubbard models also when affected by a phase problem due to an applied magnetic field.

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Berezin integrals and Poisson processes

Cited by 20 publications

References 11 publications

Renormalization group and probability theory

Renormalization group and probability theory

The exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

A perturbative probabilistic approach to quantum many-body systems

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