Random matrices: Universality of ESDs and the circular law

Tao, Terence; Vu, Van; Krishnapur, Manjunath

doi:10.1214/10-aop534

Cited by 380 publications

(531 citation statements)

References 24 publications

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“…As long as the five critical quantities are the same, two matrices whose coefficients are sampled from two different distribution will yield approximately the same eigenvalue distribution. This phenomenon is known in the random matrix theory literature as "universality" (Tao et al, 2010;Naumov, 2012).…”

Section: Discussionmentioning

confidence: 99%

Reactivity and stability of large ecosystems

Tang

Allesina

2014

Front. Ecol. Evol.

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The study of local stability has a long tradition in community ecology. Stability describes whether an ecological system will eventually return to its original steady state after being perturbed. More recently, the study of the transient dynamics of ecological systems has been recognized as crucial, given that continuously disturbed systems might never reach a steady state, and thus the instantaneous response to perturbations could largely determine species persistence. A stable equilibrium can be non-reactive-all perturbations decay immediately, or reactive-some perturbations are initially amplified before decaying. Here we derive analytical criteria for the reactivity of large ecological systems in which species interact at random. We find that in large ecological systems both stability and reactivity are governed by the same quantities: number of species, means of the intra-and inter-specific interaction strengths, variance of inter-specific interactions, and the correlation of pairwise interactions. We identify two phase transitions, one from non-reactivity to reactivity and one from stability to instability. As reactivity is an intermediate state between non-reactivity and instability, it could be used to develop an early-warning signal for systems approaching instability.

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Section: Discussionmentioning

confidence: 99%

Reactivity and stability of large ecosystems

Tang

Allesina

2014

Front. Ecol. Evol.

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“…For non-Gaussian entries whose law possesses a density and finite moments of order at least 6, a full proof, based on Girko idea's, appears in [Bai97]. The problem was recently settled in full generality, see [TaV08a], [TaV08b], [GoT07]; the extra ingredients in the proof are closely related to the study of the minimal singular value of XX * discussed above.…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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“…, y ln,n be the zeros of P ′ n located in the disk D R . Note that k n ≤ n and l n < n. By the Poisson-Jensen formula, see [6,Chapter 8], we have for any z ∈ D R which is not a zero or pole of L n , (11) log…”

Section: Proof Of Lemma 22mentioning

confidence: 99%

“…Applying the inequality between the arithmetic and quadratic means several times to the Poisson-Jensen formula (11) and dividing by n 2 we obtain 1 n 2 log 2 |L n (z)| (16) ≤ 3 n 2 I 2 n (z; R) + 3l n n 2 ln l=1 log 2 R(z − y l,n ) R 2 −ȳ l,n z + 3k n n 2 kn k=1 log 2 R(z − x k,n ) R 2 −x k,n z .…”

Section: Proof Of Lemma 22mentioning

confidence: 99%

Critical points of random polynomials with independent identically distributed roots

Kabluchko

2014

Proc. Amer. Math. Soc.

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Abstract. Let X 1 , X 2 , . . . be independent identically distributed random variables with values in C. Denote by µ the probability distribution of X 1 . Consider a random polynomial Pn(z) = (z − X 1 ) . . . (z − Xn). We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure µn := 1 n−1 P ′ n (z)=0 δz counting the complex zeros of the derivative P ′ n converges in probability to µ, as n → ∞.

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Random matrices: Universality of ESDs and the circular law

Cited by 380 publications

References 24 publications

Reactivity and stability of large ecosystems

Reactivity and stability of large ecosystems

An Introduction to Random Matrices

Critical points of random polynomials with independent identically distributed roots

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