We consider nearest–neighbor random walks on free products of finitely many copies of the integers with amalgamation over nontrivial subgroups. When all the subgroups have index two, we find the Green function of the random walks in terms of complete elliptic integrals. Our technique is to apply Voiculescu’s operator–valued R–transform.

It is shown that the positive first-order theory of an HNN-extension G = 〈H, t; tat = φ(a)(a ∈ A)〉 can be reduced to the existential firstorder theory of G provided that A and B are proper subgroups of the base group H with A∩B finite. For an amalgamated free product G = H ∗A J , we show that the positive first-order theory of G can be reduced to the existential first-order theory of G in case ...

Abstract We prove the existence of joint limiting spectral distributions for families random sample covariance matrices modeled on fluctuations discretized Lévy processes. These models were first considered in applications matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When underlying process is non-Gaussian, we show that are distinct fr...