# Singularity of Random Matrices

In this mini-course we will introduce some of the tools that are used to study the singularity probability of random matrices. We will focus on the following two fundamental models: let Bn be chosen uniformly at random from all n × n matrices with ±1 entries, and let An be chosen uniformly at random from all n × n symmetric matrices with ±1 entries.
Class 1: We will introduce the Littlewood–Offord problem, and see how anticoncentration relates to the problem of singularity. We will then discuss inverse Littlewood–Offord theorems, introduce the geometric method of Rudelson and Vershynin, and use it to prove an exponential upper bound on the singularity probability of Bn.
Class 2: We will explain why the method used in Class 1 cannot be easily extended to the setting of symmetric random matrices. We will then prove an inverse Littlewood–Offord theorem in Zp and use it to show that P(det(An) = 0) ⩽ exp(−c√n).
Class 3: We will introduce a technique that we call “inversion of randomness” and see how it can be used to deal with certain correlations in the symmetric case. We will also use Fourier analysis to prove a negative correlation inequality between very unlikely events.
Finally we will use these techniques to show that P(det(An) = 0) ⩽ exp(−cn).
Class 4: We will study the least singular value of a random symmetric matrix. We will discuss some of the new challenges that this problem poses, and see how the methods presented in the previous classes can be used to attack it.

References:
[1] M. Campos, M. Jenssen, M. Michelen and J. Sahasrabudhe, Singularity of random symmetric matrices revisited, Proc. Amer. Math. Soc., 150 (2022), 3147–3159.
[2] M. Campos, M. Jenssen, M. Michelen and J. Sahasrabudhe, The singularity probability of a random symmetric matrix is exponentially small, arXiv preprint arXiv:2105.11384.
[3] M. Campos, M. Jenssen, M. Michelen and J. Sahasrabudhe, The least singular value of a random symmetric matrix, arXiv preprint arXiv:2203.06141.
[4] M. Campos, L. Mattos, R. Morris and N. Morrison, On the singularity of random symmetric matrices, Duke Math. J., 170 (2021), 881–907.
[5] K. P. Costello, T. Tao and V. Vu, Random symmetric matrices are almost surely nonsingular, Duke Math. J., 135 (2006), 395–413.
[6] A. Ferber and V. Jain, Singularity of random symmetric matrices – a combinatorial approach to improved bounds, Forum Math. Sigma, 7 (2019).
[7] J. Kahn, J. Koml´os and E. Szemer´edi, On the probability that a random ±1 matrix is singular, J. Amer. Math. Soc., 8 (1995), 223–240.
[8] M. Rudelson and R. Vershynin, The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218 (2008), 600–633.
[9] T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, 2006.
[10] K. Tikhomirov, Singularity of random Bernoulli matrices, Ann. Math., 191 (2020), 593–634.
[11] R. Vershynin, Invertibility of symmetric random matrices, Random Struc. Alg., 44 (2014), 135–182.