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We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$, its variance, and the spectrum of the graph. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein and Bennett-type inequalities, as well as an inequality for subgaussian variables., Comment: V4: published version; theorems 1&2 slightly revised V3: Improved Bennett inequality V2: Corrected version. Confusion concerning definition of $\beta$ resolved. Results given both in terms of sepctral gap and second largest absolute value of an eigenvalue. Sign error in statement of Theorem 4 corrected. Other minor and cosmetic corrections included as well

Let M be a closed symplectic manifold, and let | | be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically., Comment: Latex, 17 pages

For an asymptotic $\ell_1$ space $X$ with a basis $(x_i)$ certain asymptotic $\ell_1$ constants, $\delta_\alpha (X)$ are defined for $\alpha <\omega_1$. $\delta_\alpha (X)$ measures the equivalence between all normalized block bases $(y_i)_{i=1}^k$ of $(x_i)$ which are $S_\alpha$-admissible with respect to $(x_i)$ ($S_\alpha$ is the $\alpha^{th}$-Schreier class of sets) and the unit vector basis of $\ell_1^k$. This leads to the concept of the delta spectrum of $X$, $\Delta (X)$, which reflects the behavior of stabilized limits of $\delta_\alpha (X)$. The analogues of these constants under all renormings of $X$ are also defined and studied. We investigate $\Delta (X)$ both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelson's space $T$ and its relatives.