Off-critical lattice models and massive SLEs

We suggest how versions of Schramm’s SLE can be used to describe the scaling limit of some off-critical 2D lattice models. Many open questions remain

Regularity of the Density of Surface States

We prove that the integrated density of surface states of continuous or discrete Anderson-type random Schroedinger operators is a measurable locally integrable function rather than a signed measure or a distribution. This generalize our recent results on the existence of the integrated density of surface states in the continuous case and those of A. Chahrour in the discrete case. The proof uses the new $L^p$-bound on the spectral shift function recently obtained by Combes, Hislop, and Nakamura. Also we provide a simple proof of their result on the Hoelder continuity of the integrated density of bulk states

The three smallest compact arithmetic hyperbolic 5-orbifolds

We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.Comment: 11 page

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all \b>0 there exists \rho (d,\b)N(d,\b), any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ | \{y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as \b \to \infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne