Finite range Decomposition of Gaussian Processes

Let \D be the finite difference Laplacian associated to the lattice \bZ^{d}. For dimension $d\ge 3$, $a\ge 0$ and $L$ a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G^{a}:=(a-\D)^{-1} can be decomposed as an infinite sum of positive semi-definite functions $V_{n}$ of finite range, $V_{n} (x-y) = 0$ for $|x-y|\ge O(L)^{n}$. Equivalently, the Gaussian process on the lattice with covariance $G^{a}$ admits a decomposition into independent Gaussian processes with finite range covariances. For $a=0$, $V_{n}$ has a limiting scaling form $L^{-n(d-2)}\Gamma_{c,\ast}{\bigl (\frac{x-y}{L^{n}}\bigr)}$ as $n\to \infty$. As a corollary, such decompositions also exist for fractional powers (-\D)^{-\alpha/2}, $0<\alpha \leq 2$. The results of this paper give an alternative to the block spin renormalization group on the lattice.Comment: 26 pages, LaTeX, paper in honour of G.Jona-Lasinio.Typos corrected, corrections in section 5 and appendix

American Indian Activism and the Rise of Red Power

Recent historical scholarship has determined that the socio-political environment of post-World War II America provided the necessary catalyst for Native American activism which when combined with the socio-political atmosphere of the civil rights era lead to the development of the Red Power Movement. In the thirty or so years immediately following World War II America witnessed profound social and political change. Initial fear of communism lead to strict, pro-capitalist Indian legislation resulting in the termination of hundreds of tribes and the relocation of countless Indian people. From this same environment rose strong leaders, including many veterans, influenced by Cold War foreign policy and American idealism. They created the foundation for Native activism as it was defined in the 1950s and 1960s. From this starting point arose native youth movements, influenced by other national youth movements, urbanization and the creation of a pan-Indian identity, and the Civil Rights era. The youth movements maintained the notion of urgency in their actions and opted for direct action as the best form of activism. The concept of Red Power accompanied activist militancy and resulted in a series of violent actions. The ideals and goals of these movements remained the same regardless of the methods used to obtain them. Self-determination, upholding of treaty rights and strong tribal sovereignty were the main fighting points for all Native activists. This new perspective on Native American activism contributes to the scholarship by first focusing on the outside environmental influences on Native activists while simultaneously seeking to clarify the movements by separating Red Power from the broader, often earlier, concept of Native activism