Fermionic construction of tau functions and random processes

Harnad, J.; Orlov, A. Yu.

doi:10.1016/j.physd.2007.05.011

Cited by 14 publications

(23 citation statements)

References 52 publications

(142 reference statements)

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“…We will claim in Section 6 that these three determinantal processes (one finite and two infinite systems) and others reported in references [78,46,95,96,51,57] have a common structure; the matrix-kernels are expressed by spectral projections associated with appropriate self-adjoint operators (effective Hamiltonians) [78]. As explained by Spohn [88] and by Prähofer and Spohn [78], this common feature is shared also with the 1+1 dimensional Fermi field in quantum mechanics (see also [35,43]). …”

Section: Matrix-kernels and Determinantal Processesmentioning

confidence: 82%

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Noncolliding Brownian Motion and Determinantal Processes

2007

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A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

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Section: Matrix-kernels and Determinantal Processesmentioning

confidence: 82%

“…The Schur function expansion is a special case of character expansions (see [8,9,59,55,43]). Let Γ(a) = ∞ 0 e −y y a−1 dy for a > 0 (the Gamma function) and note Γ(a + 1) = a!…”

Section: Schur Function Expansionmentioning

confidence: 99%

Noncolliding Brownian Motion and Determinantal Processes

2007

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“…. 44,45 Recently, the relationship between the stochastic processes such as the random turn walk and soliton theory has been discussed in Ref. It would be interesting to investigate the limiting behavior of the correlation kernel in a more general situation and how it depends on these parameters.…”

Section: Discussionmentioning

confidence: 99%

Correlation function of the Schur process with a fixed final partition

Imamura

Sasamoto

2008

Journal of Mathematical Physics

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We consider a generalization of the Schur process in which a partition evolves from the empty partition into an arbitrary fixed final partition. We obtain a double integral representation of the correlation kernel. For a special final partition with only one row, the edge scaling limit is also discussed by the use of the saddle point analysis. If we appropriately scale the length of the row, the limiting correlation kernel changes from the extended Airy kernel.Comment: 28 pages, 2 figure

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“…We are thankful to John Harnad for kind hospitality and numerous fruitful discussions which allowed to create this paper which may be viewed as a continuation of [4] and [11]. We thank Marco Bertolla and other participants of the working seminar on Integrable Systems, Random Matrices and Random Processes in Concordia university headed by J. Harnad for interesting…”

Section: Acknowledgementsmentioning

confidence: 99%

“…However in that case a quadric fermionic Hamiltoinian is used to describe the stochastic dynamics of particles which was identified with Hamiltonian dynamics of quantum spin (or, of nonlinear fermionic) system where the (real) wave function yields the probability distribution for the stochastic process. Following [4] we use free fermions and the answer for the probability is given by the ratio of two factors (35) quite similar to what we have in thermodynamics where the probability of a state is the ratio of the weight of the state and a normalization function (the partition function).…”

Section: Introductionmentioning

confidence: 99%

Random turn walk on a half line with creation of particles at the origin

Leur¹,

Orlov²

2009

Physics Letters A

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We consider a version of random motion of hard core particles on the semi-lattice 1, 2, 3, . . ., where in each time instant one of three possible events occurs, viz., (a) a randomly chosen particle hops to a free neighboring site, (b) a particle is created at the origin (namely, at site 1) provided that site 1 is free and (c) a particle is eliminated at the origin (provided that the site 1 is occupied). Relations to the BKP equation are explained. Namely, the tau functions of two different BKP hierarchies provide generating functions respectively (I) for transition weights between different particle configurations and (II) for an important object: a normalization function which plays the role of the statistical sum for our non-equilibrium system. As an example we study a model where the hopping rate depends on two parameters (r and β). For time T → ∞ we obtain the asymptotic configuration of particles obtained from the initial empty state (the state without particles) and find an analog of the first order transition at β = 1.

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Fermionic construction of tau functions and random processes

Cited by 14 publications

References 52 publications

Noncolliding Brownian Motion and Determinantal Processes

Noncolliding Brownian Motion and Determinantal Processes

Correlation function of the Schur process with a fixed final partition

Random turn walk on a half line with creation of particles at the origin

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