Thin-shell theory for rotationally invariant random simplices

For fixed functions G,H:[0,∞)→[0,∞), consider the rotationally invariant probability density on Rn of form μn(ds)=1 ZnG(‖s‖2)e−nH(‖s‖2)ds. We show that when n is large, Euclidean norm ‖Yn‖2 a random vector Yn distributed according to μn satisfies thin-shell property, in its distribution highly likely concentrate around value s0 minimizing certain variational problem. Moreover, we fluctuations this modulus away from have order 1∕n and are approximately Gaussian large. apply these observations simplices: simplex whose vertices consist origin as well independent vectors Y1n,…,Ypn μn, ultimately showing logarithmic volume resulting exhibits behavior. Our class measures includes distribution, beta prime Rn, provided generalizing unifying setting for objects considered Grote-Kabluchko-Thäle [Limit theorems simplices high dimensions, ALEA 16, 141–177 (2019)]. Finally, volumes may be related determinants matrices, use our methods with correspondence if An an n×n matrix entries standard variables, then there explicit constants c0,c1∈(0,∞) absolute constant C∈(0,∞) such sups∈RPlogdet(An)−log(n−1)!−c0 1 2logn+c1<s−∫−∞se−u2∕2du 2π<C log3∕2n, sharpening 1∕log1∕3+o(1)n bound Nguyen Vu [Random matrices: Law determinant, Ann. Probab. 42(1) (2014), 146–167].