The smallest singular value of inhomogeneous square random matrices

We show that, for an n×n random matrix A with independent uniformly anticoncentrated entries such that E‖A‖HS2≤Kn2, the smallest singular value σn(A) of satisfies P{σn(A)≤ε n}≤Cε+2e−cn,ε≥0. This extends earlier results (Adv. Math. 218 (2008) 600–633; Israel J. 227 (2018) 507–544) by removing assumption mean zero and identical distribution across as well recent result (Livshyts (2018)) where was required to have i.i.d. rows. Our model covers inhomogeneous matrices allowing different variances long sum second moments is order O(n2). In past advances, rows due lack Littlewood–Offord-type inequalities weighted sums non-i.i.d. variables. Here, we overcome this problem introducing Randomized Least Common Denominator (RLCD) which allows study anti-concentration properties but not identically distributed construct efficient nets on sphere lattice structure points typically large RLCD. us derive strong anticoncentration distance between a fixed column linear span remaining columns prove main result.