82 research outputs found
Strong games played on random graphs
In a strong game played on the edge set of a graph G there are two players,
Red and Blue, alternating turns in claiming previously unclaimed edges of G
(with Red playing first). The winner is the first one to claim all the edges of
some target structure (such as a clique, a perfect matching, a Hamilton cycle,
etc.). It is well known that Red can always ensure at least a draw in any
strong game, but finding explicit winning strategies is a difficult and a quite
rare task. We consider strong games played on the edge set of a random graph G
~ G(n,p) on n vertices. We prove, for sufficiently large and a fixed
constant 0 < p < 1, that Red can w.h.p win the perfect matching game on a
random graph G ~ G(n,p)
Singularity of random symmetric matrices -- a combinatorial approach to improved bounds
Let denote a random symmetric matrix whose upper diagonal
entries are independent and identically distributed Bernoulli random variables
(which take values and with probability each). It is widely
conjectured that is singular with probability at most . On
the other hand, the best known upper bound on the singularity probability of
, due to Vershynin (2011), is , for some unspecified small
constant . This improves on a polynomial singularity bound due to
Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the
singularity probability decays faster than any polynomial. In this paper,
improving on all previous results, we show that the probability of singularity
of is at most for all sufficiently
large . The proof utilizes and extends a novel combinatorial approach to
discrete random matrix theory, which has been recently introduced by the
authors together with Luh and Samotij.Comment: Final version incorporating referee comment
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