The theory of random matrices contains many central limit theorems. We have central limit theorem... more The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and many others. In this notes, we discuss the following problem: Is it possible to prove the law of the iterated logarithm? We illustrate this possibility by showing that this is indeed the case for the log of the permanent of random Bernoulli matrices and pose open questions concerning several other matrix parameters.
A community of $n$ individuals splits into two camps, Red and Blue. The individuals are connected... more A community of $n$ individuals splits into two camps, Red and Blue. The individuals are connected by a social network, which influences their colors. Everyday, each person changes his/her color according to the majority among his/her neighbors. Red (Blue) wins if everyone in the community becomes Red (Blue) at some point. We study this process when the underlying network is the random Erdos-Renyi graph $G(n, p)$. With a balanced initial state ($n/2$ person in each camp), it is clear that each color wins with the same probability. Our study reveals that for any constants $p$ and $\varepsilon$, there is a constant $C$ such that if one camp has $n/2 +C$ individuals, then it wins with probability at least $1 - \varepsilon$. The surprising key fact here is that $C$ does not depend on $n$, the population of the community. When $p=1/2$ and $\varepsilon =.1$, one can set $C$ as small as 6. If the aim of the process is to choose a candidate, then this means it takes only $6$ "defectors&...
We introduce a new procedure for generating the binomial random graph/hypergraph models, referred... more We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer k≥3 , the binomial k ‐uniform random hypergraph Hn,pk contains N:=(1−o(1))(n−1k−1)p edge‐disjoint perfect matchings, provided p≥logCnnk−1 , where C:=C(k) is an integer depending only on k . Our result for N is asymptotically optimal and for p is optimal up to the polylog(n) factor. This significantly improves a result of Frieze and Krivelevich.
In this paper, we prove optimal local universality for roots of random polynomials with arbitrary... more In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.
In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic b... more In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic block model with $k$ blocks, for any $k$ fixed. Our algorithm works with graphs having constant edge density, under an optimal condition on the gap between the density inside a block and the density between the blocks. As a co-product, we settle an open question posed by Abbe et. al. concerning censor block models.
Roots of random polynomials have been studied intensively in both analysis and probability for a ... more Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős–Offord, showed that the expectation of the number of real roots is [Formula: see text]. In this paper, we determine the true nature of the error term by showing that the expectation equals [Formula: see text]. Prior to this paper, the error term [Formula: see text] has been known only for polynomials with Gaussian coefficients.
The theory of random matrices contains many central limit theorems. We have central limit theorem... more The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and many others. In this notes, we discuss the following problem: Is it possible to prove the law of the iterated logarithm? We illustrate this possibility by showing that this is indeed the case for the log of the permanent of random Bernoulli matrices and pose open questions concerning several other matrix parameters.
A community of $n$ individuals splits into two camps, Red and Blue. The individuals are connected... more A community of $n$ individuals splits into two camps, Red and Blue. The individuals are connected by a social network, which influences their colors. Everyday, each person changes his/her color according to the majority among his/her neighbors. Red (Blue) wins if everyone in the community becomes Red (Blue) at some point. We study this process when the underlying network is the random Erdos-Renyi graph $G(n, p)$. With a balanced initial state ($n/2$ person in each camp), it is clear that each color wins with the same probability. Our study reveals that for any constants $p$ and $\varepsilon$, there is a constant $C$ such that if one camp has $n/2 +C$ individuals, then it wins with probability at least $1 - \varepsilon$. The surprising key fact here is that $C$ does not depend on $n$, the population of the community. When $p=1/2$ and $\varepsilon =.1$, one can set $C$ as small as 6. If the aim of the process is to choose a candidate, then this means it takes only $6$ "defectors&...
We introduce a new procedure for generating the binomial random graph/hypergraph models, referred... more We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer k≥3 , the binomial k ‐uniform random hypergraph Hn,pk contains N:=(1−o(1))(n−1k−1)p edge‐disjoint perfect matchings, provided p≥logCnnk−1 , where C:=C(k) is an integer depending only on k . Our result for N is asymptotically optimal and for p is optimal up to the polylog(n) factor. This significantly improves a result of Frieze and Krivelevich.
In this paper, we prove optimal local universality for roots of random polynomials with arbitrary... more In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.
In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic b... more In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic block model with $k$ blocks, for any $k$ fixed. Our algorithm works with graphs having constant edge density, under an optimal condition on the gap between the density inside a block and the density between the blocks. As a co-product, we settle an open question posed by Abbe et. al. concerning censor block models.
Roots of random polynomials have been studied intensively in both analysis and probability for a ... more Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős–Offord, showed that the expectation of the number of real roots is [Formula: see text]. In this paper, we determine the true nature of the error term by showing that the expectation equals [Formula: see text]. Prior to this paper, the error term [Formula: see text] has been known only for polynomials with Gaussian coefficients.
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