Local Semicircle Law Under Fourth Moment Condition

We consider a random symmetric matrix $$\mathbf{X}= [X_{jk}]_{j,k=1}^n$$ X = [ X jk ] j , k = 1 n with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $$ \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4+\delta } 0$$ max jk E | X jk | 4 + δ 0 , it was proved in Götze et al. (Bernoulli 24(3):2358–2400, 2018) that with high probability the typical distance between the Stieltjes transforms $$m_n(z)$$ m n ( z ) , $$z = u + i v$$ z = u + i v , of the empirical spectral distribution (ESD) and the Stieltjes transforms $$m_{\text {sc}}(z)$$ m sc ( z ) of the semicircle law is of order $$(nv)^{-1} \log n$$ ( n v ) - 1 log n . The aim of this paper is to remove $$\delta >0$$ δ > 0 and show that this result still holds if we assume that $$ \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4}