Non-intersecting Brownian Bridges in the Flat-to-Flat Geometry

Grela, Jacek; Majumdar, Satya N.; Schehr, Grégory

doi:10.1007/s10955-021-02774-6

Cited by 26 publications

(30 citation statements)

References 49 publications

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“…In that case there is an additional repulsive two-body interaction δW on top of the interaction (84), of the form δW (x, y) ∝ −aβ (x − y) coth 1 2 (x − y), which never vanishes for any value of β (the model is always interacting). This model is related to the Stieltjes-Wigert βensemble (SWβE) [118,[126][127][128][129] which was studied in the context of Chern-Simons theory in high energy physics [130] and of non-intersecting Brownian bridges [129,131,132]. The correspondence is through the map (see the discussion below Eq.…”

Section: Tablementioning

confidence: 99%

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Full counting statistics for interacting trapped fermions

et al. 2021

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We study NN spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index \betaβ. In the fermion model \betaβ controls the strength of the interaction, \beta=2β=2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_DND in a domain DD of macroscopic size in the bulk of the Fermi gas. We predict that for general \betaβ the variance of N_DND grows as A_{\beta} \log N + B_{\beta}AβlogN+Bβ for N \gg 1N≫1 and we obtain a formula for A_\betaAβ and B_\betaBβ. This is based on an explicit calculation for \beta\in\left\{ 1,2,4\right\}β∈{1,2,4} and on a conjecture that we formulate for general \betaβ. This conjecture further allows us to obtain a universal formula for the higher cumulants of N_DND. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/\betaK=2/β, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter \betaβ. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

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Section: Tablementioning

confidence: 99%

“…For this matrix model the joint PDF of the eigenvalues is determinantal for β = 2, since the model becomes bi-orthogonal [133][134][135]. In the limit of large N , scaling a = O(N ), the eigenvalue density σ(λ) is known [129,130,132]…”

Section: Tablementioning

confidence: 99%

Full counting statistics for interacting trapped fermions

et al. 2021

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“…In most cases, this quantity is not known analytically but when it is at our disposal, Doob's technique has been successfully applied to various kinds of conditioned processes [1,11,[15][16][17]. This approach continues to attract intense research efforts, mostly aimed at extending the range of applicability of the Doob theory, for instance towards discrete-time constrained random walks and Lévy flights [18,19], run-and-tumble trajectories [20], processes with resetting [21], or non-intersecting Brownian bridges [22].…”

Section: Introductionmentioning

confidence: 99%

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Monthus¹,

Mazzolo²

2022

Preprint

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When the unconditioned process is a diffusion living on the half-line x ∈] − ∞, a[ in the presence of an absorbing boundary condition at position x = a, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite T < +∞, the conditioning consists in imposing the probability P * (y, T ) to be surviving at time T and at the position y ∈ ] − ∞, a[, as well as the probability γ * (Ta) to have been absorbed at the previous time Ta ∈ [0, T ]. When the time horizon is infinite T = +∞, the conditioning consists in imposing the probability γ * (Ta) to have been absorbed at the time Ta ∈ [0, +∞[, whose normalization [1−S * (∞)] determines the conditioned probability S * (∞) ∈ [0, 1] of forever-survival. This case of infinite horizon T = +∞ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passagetime properties at position a. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift µ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.

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“…Yet another interpretation in this case is that the eigenvalues of M follow the same joint probability distribution as n non-intersecting Brownian motions with the eigenvalues of A as starting points and 0 as the common endpoint [30,31].…”

Section: Introductionmentioning

confidence: 99%

“…Hence, the special case of (1.6) with f (x) = x θ is also called the Muttalib-Borodin ensemble, see [29,36,38,20,19] for example. Biorthogonal ensembles and variations thereof are also considered in [5,37,14,17,22,24,9,8,30,31].…”

Section: Introductionmentioning

confidence: 99%

Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemble

Claeys,

Wang

2022

Preprint

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We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel-Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogonal ensembles in general.

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Non-intersecting Brownian Bridges in the Flat-to-Flat Geometry

Cited by 26 publications

References 49 publications

Full counting statistics for interacting trapped fermions

Full counting statistics for interacting trapped fermions

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemble

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