The Traffic Distribution of the Squared Unimodular Random Matrix and a Formula for the Moments of its ESD

Jorge Garza Vargas

doi:10.1016/S0034-4877(18)30044-2

The Traffic Distribution of the Squared Unimodular Random Matrix and a Formula for the Moments of its ESD

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Abstract

The k-th moment of the mean empirical spectral distribution (ESD) of the squared unimodular random matrix of dimension N can be expressed in the form N^−2k—1Q_k(N), where Q_k(x) is a polynomial of degree k + 1 with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Puchała, K. Życzkowski [3].

Original language	English (US)
Pages (from-to)	273-282
Number of pages	10
Journal	Reports on Mathematical Physics
Volume	81
Issue number	3
DOIs	https://doi.org/10.1016/S0034-4877(18)30044-2
State	Published - Jun 2018
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

Keywords

non-commutative probability
quantum information theory
random matrices

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10.1016/S0034-4877(18)30044-2

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title = "The Traffic Distribution of the Squared Unimodular Random Matrix and a Formula for the Moments of its ESD",

abstract = "The k-th moment of the mean empirical spectral distribution (ESD) of the squared unimodular random matrix of dimension N can be expressed in the form N−2k—1Qk(N), where Qk(x) is a polynomial of degree k + 1 with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Pucha{\l}a, K. {\.Z}yczkowski [3].",

keywords = "non-commutative probability, quantum information theory, random matrices",

author = "Vargas, \{Jorge Garza\}",

note = "Publisher Copyright: {\textcopyright} 2018 Polish Scientific Publishers",

year = "2018",

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doi = "10.1016/S0034-4877(18)30044-2",

language = "English (US)",

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journal = "Reports on Mathematical Physics",

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T1 - The Traffic Distribution of the Squared Unimodular Random Matrix and a Formula for the Moments of its ESD

AU - Vargas, Jorge Garza

PY - 2018/6

Y1 - 2018/6

N2 - The k-th moment of the mean empirical spectral distribution (ESD) of the squared unimodular random matrix of dimension N can be expressed in the form N−2k—1Qk(N), where Qk(x) is a polynomial of degree k + 1 with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Puchała, K. Życzkowski [3].

AB - The k-th moment of the mean empirical spectral distribution (ESD) of the squared unimodular random matrix of dimension N can be expressed in the form N−2k—1Qk(N), where Qk(x) is a polynomial of degree k + 1 with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Puchała, K. Życzkowski [3].

KW - non-commutative probability

KW - quantum information theory

KW - random matrices

UR - https://www.scopus.com/pages/publications/85049088386

UR - https://www.scopus.com/inward/citedby.url?scp=85049088386&partnerID=8YFLogxK

U2 - 10.1016/S0034-4877(18)30044-2

DO - 10.1016/S0034-4877(18)30044-2

M3 - Article

AN - SCOPUS:85049088386

SN - 0034-4877

VL - 81

SP - 273

EP - 282

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

IS - 3

ER -

The Traffic Distribution of the Squared Unimodular Random Matrix and a Formula for the Moments of its ESD

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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