Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions

Forrester, Peter J.

doi:10.1088/0305-4470/39/41/s14

Cited by 9 publications

(18 citation statements)

References 17 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…He assumed that a uniform background with negative charge density −N/LW exists and the system is neutralized. Such a system consisting of positively charged N particles and negatively charged background is called a one-component plasma model [20,12,13]. In the present paper, we show that the elliptic DPP of type A N −1 is identified with the particle section of Forrester's one-component plasma model, in which the background effect is subtracted.…”

Section: Introductionmentioning

confidence: 52%

“…where ρ = N/(LW ). As pointed out by Forrester [12], it was shown by Cardy [9] that, if we ignore the zero mode, the partition function of the Gaussian free field (GFF) on a torus with the modular parameter τ ∈ H is given by…”

Section: Relationship To Gaussian Free Field On a Torusmentioning

confidence: 99%

“…The former system was studied by Forrester [12], but the latter is new. The total energies of the particleparticle interaction in particle configurations z are given by…”

Section: General Setting Of One-component Plasma Modelsmentioning

confidence: 99%

“…First we review the argument given by Forrester [12] for the first system with the two-point potential function Φ − . In (4.10), put…”

Section: Solvability Conditionsmentioning

confidence: 99%

“…where s(N ) = 0, if N is even, and s(N ) = 1, if N is odd. We can prove the following identities [12]. (The proof is given in Appendix C.) (4.13)…”

Section: Solvability Conditionsmentioning

confidence: 99%

See 4 more Smart Citations

Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

Katori

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

We introduce new families of determinantal point processes (DPPs) on a complex plane C, which are classified into seven types following the irreducible reduced affine root systems,Their multivariate probability densities are doubly periodic with periods (L, iW ), 0 < L, W < ∞, i = √ −1. The construction is based on the orthogonality relations with respect to the double integrals over the fundamental domain, [0, L) × i[0, W ), which are proved in this paper for the RN -theta functions introduced by Rosengren and Schlosser. In the scaling limit N → ∞, L → ∞ with constant density ρ = N/(LW ) and constant W , we obtain four types of DPPs with an infinite number of points on C, which have periodicity with period iW . In the further limit W → ∞ with constant ρ, they are degenerated into three infinite-dimensional DPPs. One of them is uniform on C and equivalent with the Ginibre point process studied in random matrix theory, while other two systems are rotationally symmetric around the origin, but non-uniform on C. We show that the elliptic DPP of type AN−1 is identified with the particle section, obtained by subtracting the background effect, of the two-dimensional exactly solvable model for one-component plasma studied by Forrester. Other two exactly solvable models of one-component plasma are constructed associated with the elliptic DPPs of types CN and DN . Relationship to the Gaussian free field on a torus is discussed for these three exactly solvable plasma models. *

show abstract

Section: Introductionmentioning

confidence: 52%

Section: Relationship To Gaussian Free Field On a Torusmentioning

confidence: 99%

“…The former system was studied by Forrester [12], but the latter is new. The total energies of the particleparticle interaction in particle configurations z are given by…”

Section: General Setting Of One-component Plasma Modelsmentioning

confidence: 99%

“…First we review the argument given by Forrester [12] for the first system with the two-point potential function Φ − . In (4.10), put…”

Section: Solvability Conditionsmentioning

confidence: 99%

“…where s(N ) = 0, if N is even, and s(N ) = 1, if N is odd. We can prove the following identities [12]. (The proof is given in Appendix C.) (4.13)…”

Section: Solvability Conditionsmentioning

confidence: 99%

See 3 more Smart Citations

Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

Katori

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

Berman

2018

Algebraic and Analytic Microlocal Analysis

View full text Add to dashboard Cite

We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Ampère operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials. 7 9 42

show abstract

Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes

Katori

Shirai

2022

Commun. Math. Phys.

View full text Add to dashboard Cite

On an annulus $${{\mathbb {A}}}_q :=\{z \in {{\mathbb {C}}}: q< |z| < 1\}$$ A q : = { z ∈ C : q < | z | < 1 } with a fixed $$q \in (0, 1)$$ q ∈ ( 0 , 1 ) , we study a Gaussian analytic function (GAF) and its zero set which defines a point process on $${{\mathbb {A}}}_q$$ A q called the zero point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by $$r >0$$ r > 0 is identified with the weighted Szegő kernel of $${{\mathbb {A}}}_q$$ A q with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate $$z \leftrightarrow q/z$$ z ↔ q / z and the parameter change $$r \leftrightarrow q^2/r$$ r ↔ q 2 / r . When $$r=q$$ r = q they are invariant under conformal transformations which preserve $${{\mathbb {A}}}_q$$ A q . Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of McCullough and Shen but the weight parameter r is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit $$q \rightarrow 0$$ q → 0 , a simpler but still non-trivial PDPP is obtained on the unit disk $${\mathbb {D}}$$ D . We observe that the limit PDPP indexed by $$r \in (0, \infty )$$ r ∈ ( 0 , ∞ ) can be regarded as an interpolation between the determinantal point process (DPP) on $${{\mathbb {D}}}$$ D studied by Peres and Virág ($$r \rightarrow 0$$ r → 0 ) and that DPP of Peres and Virág with a deterministic zero added at the origin ($$r \rightarrow \infty $$ r → ∞ ).

show abstract

Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions

Cited by 9 publications

References 17 publications

Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes

Contact Info

Product

Resources

About