For every \\( {A\\in\\mathcal{M}_n(\\mathbb{C})} \\) let us define<\/p>\n
\\[ s(A):=\\min_{\\Vert x\\Vert_2=1}\\Vert Ax\\Vert_2 \\quad\\text{and}\\quad \\Vert A\\Vert:=\\max_{\\Vert x\\Vert_2=1}\\Vert Ax\\Vert_2. \\]<\/p>\n
Let \\( {X} \\) be a random matrix in \\( {\\mathcal{M}_n(\\mathbb{C})} \\) with i.i.d. entries of mean \\( {m:=\\mathbb{E}(X_{11})} \\) and unit variance. Fix \\( {0<s_-\\leq s_+<\\infty} \\) and let \\( {A_1,\\ldots,A_n} \\) be invertible deterministic matrices in \\( {\\mathcal{M}_n(\\mathbb{C})} \\) s.t.<\/p>\n
\\[ s_- \\leq \\min_{1\\leq k\\leq n}s(A_k) \\leq \\max_{1\\leq k\\leq n}\\Vert A_k\\Vert\\leq s_+. \\]<\/p>\n
Let \\( {R_1,\\ldots,R_n} \\) be the rows of \\( {X} \\) and \\( {Y} \\) the random matrix with rows \\( {R_1A_1,\\ldots,R_nA_n} \\) .<\/p>\n
Conjecture (RV).<\/b> If \\( {X_{11}} \\) is sub-Gaussian, i.e. there exists \\( {c_0} \\) such that for every \\( {t\\geq0} \\),<\/p>\n
\\[ \\mathbb{P}(|X_{11}-m|>t)\\leq 2 e^{-c_0t^2} \\]<\/p>\n
then there exists \\( {C>0} \\) and \\( {c\\in(0,1)} \\) depending (polynomially) only on \\( {m} \\), \\( {c_0} \\), \\( {s_{\\pm}} \\), such that for large enough \\( {n} \\) and every \\( {\\varepsilon\\geq0} \\),<\/p>\n
\\[ \\mathbb{P}(s(Y)\\leq \\varepsilon) \\leq C\\varepsilon+c^n. \\]<\/p>\n
Conjecture (TV).<\/b> For every \\( {a>0} \\) there exists \\( {b>0} \\) depending only on \\( {a,c,m,s_{\\pm}} \\), such that for every deterministic matrix \\( {A\\in\\mathcal{M}_n(\\mathbb{C})} \\) with \\( {\\Vert A\\Vert=O(n^c)} \\) and large enough \\( {n} \\),<\/p>\n
\\[ \\mathbb{P}(s(Y+A)\\leq n^{-b}) \\leq n^{-a}. \\]<\/p>\n
These conjectures involve a transformation of \\( {X} \\), which leaves invariant the results of Adamczak et al<\/a> on the smallest singular values of random matrices with i.i.d. centered log-concave rows.<\/p>\n","protected":false},"excerpt":{"rendered":" For every \\( {A\\in\\mathcal{M}_n(\\mathbb{C})} \\) let us define \\[ s(A):=\\min_{\\Vert x\\Vert_2=1}\\Vert Ax\\Vert_2 \\quad\\text{and}\\quad \\Vert A\\Vert:=\\max_{\\Vert x\\Vert_2=1}\\Vert Ax\\Vert_2. \\] Let \\( {X} \\) be a random…<\/p>\n