scholarly journals Spectra of Random Regular Hypergraphs

10.37236/8741 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Ioana Dumitriu ◽  
Yizhe Zhu
Keyword(s):  
Random Walk ◽  
Spectral Distribution ◽  
Spectral Gap ◽  
Mixing Rate ◽  
Empirical Spectral Distribution ◽  
Expansion Property ◽  
Local Law

In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996). Our main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs. We then relate the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hypergraphs. We also prove the spectral gap for the non-backtracking operator of a random regular hypergraph introduced in Angelini et al. (2015). Finally, we obtain the convergence of the empirical spectral distribution (ESD) for random regular hypergraphs in different regimes. Under certain conditions, we can show a local law for the ESD.

Patterned sparse random matrices: A moment approach

2017 ◽  
Vol 06 (03) ◽  
pp. 1750011
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Random Matrices ◽  
Limit Distribution ◽  
Spectral Distribution ◽  
Explicit Description ◽  
Circulant Matrices ◽  
Hankel Matrices ◽  
Largest Eigenvalue ◽  
Empirical Spectral Distribution ◽  
Moment Approach ◽  
The Largest Eigenvalue

We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

scholarly journals NON-BACKTRACKING RANDOM WALKS MIX FASTER

2007 ◽  
Vol 09 (04) ◽  
pp. 585-603 ◽  
Author(s):  
NOGA ALON ◽  
ITAI BENJAMINI ◽  
EYAL LUBETZKY ◽  
SASHA SODIN
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Chebyshev Polynomials ◽  
Simple Random Walk ◽  
Mixing Rate ◽  
Ramanujan Graph ◽  
Mixing Rates

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than [Formula: see text] times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω( log n) times.

scholarly journals Circular law for random matrices with unconditional log-concave distribution

2015 ◽  
Vol 17 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Radosław Adamczak ◽  
Djalil Chafaï
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Real Matrix ◽  
Random Real ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Ginibre Ensemble ◽  
Mean And Variance ◽  
Uniform Law ◽  
Special Case

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

scholarly journals The circular law for random regular digraphs with random edge weights

2017 ◽  
Vol 06 (03) ◽  
pp. 1750012 ◽  
Author(s):  
Nicholas Cook
Keyword(s):  
Lebesgue Measure ◽  
Unit Disk ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Spectral Distribution ◽  
Hadamard Product ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Unit Variance ◽  
The Matrix

We consider random [Formula: see text] matrices of the form [Formula: see text], where [Formula: see text] is the adjacency matrix of a uniform random [Formula: see text]-regular directed graph on [Formula: see text] vertices, with [Formula: see text] for some fixed [Formula: see text], and [Formula: see text] is an [Formula: see text] matrix of i.i.d. centered random variables with unit variance and finite [Formula: see text]th moment (here ∘ denotes the matrix Hadamard product). We show that as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges weakly in probability to the normalized Lebesgue measure on the unit disk.

scholarly journals Convergence of the empirical spectral distribution function of Beta matrices

Bernoulli ◽  
2015 ◽  
Vol 21 (3) ◽  
pp. 1538-1574 ◽  
Author(s):  
Zhidong Bai ◽  
Jiang Hu ◽  
Guangming Pan ◽  
Wang Zhou
Keyword(s):  
Distribution Function ◽  
Spectral Distribution ◽  
Empirical Spectral Distribution ◽  
Spectral Distribution Function
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