Periodic Airy process and equilibrium dynamics of edge fermions in a trap

Doussal, Pierre Le; Majumdar, Satya N.; Schehr, Grégory

doi:10.1016/j.aop.2017.05.018

Cited by 26 publications

(54 citation statements)

References 50 publications

(172 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We find that, in a suitably tuned thermodynamic limit, our model provides a "crossover" between these two behaviors. The interpolating distribution depends on a positive parameter α and was previously encountered by Johansson in the so-called MNS random matrix model [Joh07], and by Le Doussal et al for free fermions in a confining trap [DLDMS16,LDMS17]-see also [LW17] and the discussion in Section 8. It is given explicitly by a Fredholm determinant F α psq :" detpI´M α q L 2 ps,8q , M α px, yq :" where Ai is the Airy function and α a positive parameter.…”

Section: Introductionmentioning

confidence: 84%

“…Interestingly, the cylindric Plancherel measure admits a stationary continuous-time periodic extension, which is the periodic analogue of the stationary process of Borodin and Olshanski [BO06b], and which we call the cylindric Plancherel process-see Figure 2. We show that its correlation kernel converges in the edge crossover regime to the extended finite-temperature Airy kernel [LDMS17]. The cylindric Plancherel process can be thought as a certain "poissonian" limit of a measure on cylindric partitions, as described below.…”

Section: Introductionmentioning

confidence: 89%

“…The limit α Ñ 0 is more subtle as it requires a rescaling. The kernel M α has been called the finite-temperature Airy kernel [LDMS17]. Our main result may then be stated as follows.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Periodic Schur Process and Free Fermions at Finite Temperature

Betea

Bouttier

2019

Math Phys Anal Geom

View full text Add to dashboard Cite

We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin's "shift-mixing" trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certain crossover regime different from that for the bulk, by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy-Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur's P and Q functions, and neutral fermions.

show abstract

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

The Periodic Schur Process and Free Fermions at Finite Temperature

Betea

Bouttier

2019

Math Phys Anal Geom

View full text Add to dashboard Cite

show abstract

“…The latter problem can be studied using a mapping to non-interacting fermions in a harmonic potential (as in Ref. [43,44]). These transformations extend in the presence of a moving barrier g(τ ) = W √ τ .…”

Section: Introductionmentioning

confidence: 99%

Non-crossing Brownian Paths and Dyson Brownian Motion Under a Moving Boundary

et al. 2019

Self Cite

View full text Add to dashboard Cite

We compute analytically the probability S(t) that a set of N Brownian paths do not cross each other and stay below a moving boundary g(τ ) = W √ τ up to time t. We show that for large t it decays as a power law S(t) ∼ t −β(N,W ) . The decay exponent β(N, W ) is obtained as the ground state energy of a quantum system of N non interacting fermions in a harmonic well in the presence of an infinite hard wall at position W . Explicit expressions for β(N, W ) are obtained in various limits of N and W , in particular for large N and large W . We obtain the joint distribution of the positions of the walkers in the presence of the moving barrier g(τ ) = W √ τ at large time. We extend our results to the case of N Dyson Brownian motions (corresponding to the Gaussian Unitary Ensemble) in the presence of the same moving boundary g(τ ) = W √ τ . For W = 0 we show that the system provides a realization of a Laguerre biorthogonal ensemble in random matrix theory. We obtain explicitly the average density near the barrier, as well as in the bulk far away from the barrier. Finally we apply our results to N non-crossing Brownian bridges on the interval [0, T ] under a time-dependent barrier gB(τ ) = W τ (1 − τ T ).

show abstract

“…This last paper has attracted much attention and it has already been cited many times in one and a half year. See, e.g., [5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Decay of Complex-Time Determinantal and Pfaffian Correlation Functionals in Lattices

Aza

Bru

Pedra

2018

Commun. Math. Phys.

View full text Add to dashboard Cite

We supplement the determinantal and Pfaffian bounds of [4] for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time correlations. Our proof uses the analyticity of correlation functions via the Hadamard three-line theorem. We show that the dynamical localization for the one-particle system yields the dynamical localization for the many-point fermionic correlation functions, with respect to the Hausdorff distance in the determinantal case. In [4], a stronger notion of decay for many-particle configurations was used but only at dimension one and for real times. Considering determinantal and pfaffian correlation functionals for complex times is important in the study of weakly interacting fermions.

show abstract

Periodic Airy process and equilibrium dynamics of edge fermions in a trap

Cited by 26 publications

References 50 publications

The Periodic Schur Process and Free Fermions at Finite Temperature

The Periodic Schur Process and Free Fermions at Finite Temperature

Non-crossing Brownian Paths and Dyson Brownian Motion Under a Moving Boundary

Decay of Complex-Time Determinantal and Pfaffian Correlation Functionals in Lattices

Contact Info

Product

Resources

About