Critical domain walls in the Ashkin-Teller model

We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE$_\kappa$, except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE$_\kappa$ in this case.Comment: Final version published in JSTAT. 13 pages, 5 figures. Substantial changes in the data production, analysis and in the conclusions. Added a section about the crossing probability. Typeset with 'iopart

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

In the introduction to this volume, we discuss some of the highlights of the research career of Chuck Newman. This introduction is divided into two main sections, the first covering Chuck's work in statistical mechanics and the second his work in percolation theory, continuum scaling limits, and related topics.Comment: 38 pages (including many references), introduction to Festschrift in honor of C.M. Newma

The Difference Between a Discrete and Continuous Harmonic Measure

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius of $h$. For a simply connected domain $D$ in the plane, let $\omega_h(0,\cdot;D)$ be the discrete harmonic measure at $0\in D$ associated with this random walk, and $\omega(0,\cdot;D)$ be the (continuous) harmonic measure at $0$. For domains $D$ with analytic boundary, we prove there is a bounded continuous function $\sigma_D(z)$ on $\partial D$ such that for functions $g$ which are in $C^{2+\alpha}(\partial D)$ for some $\alpha>0$ $$\lim_{h\downarrow 0} \frac{\int_{\partial D} g(\xi) \omega_h(0,|d\xi|;D) -\int_{\partial D} g(\xi)\omega(0,|d\xi|;D)}{h} = \int_{\partial D}g(z) \sigma_D(z) |dz|.$$ We give an explicit formula for $\sigma_D$ in terms of the conformal map from $D$ to the unit disc. The proof relies on some fine approximations of the potential kernel and Green's function of the random walk by their continuous counterparts, which may be of independent interest.Comment: 16 pages, revision after the referee's report, to appear in Journal of Theoretical Probabilit