Abstract: We consider uniform random domino tilings of the restricted Aztec diamond which is obtained by cutting off an upper triangular part of the Aztec diamond by a horizontal line. The restriction line asymptotically touches the arctic circle that is the limit shape of the north polar region in the unrestricted model. We prove that the rescaled boundary of the north polar region in the restricted domain converges to the Airy$_2$ process conditioned to stay below a parabola with explicit continuous statistics and the finite dimensional distribution kernels. The limit is the hard-edge tacnode process which was first discovered in the framework of non-intersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the non-intersecting line ensemble which corresponds to a random tiling.

Abstract: Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions which are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that $SBD^p$ functions are approximately continuous $H^n-1$-a.e. away from the jump set. On the negative side, we construct a function which is $BD$ but not in BV and has distributional strain consisting only of a jump part, and one which has a distributional strain consisting of only a Cantor part.

Abstract: We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density $\rho$ on $\mathbb{Z}_-$ and $\lambda$ on $\mathbb{Z}_+$, and a second class particle initially at the origin. For $\rho<\lambda$, there is a shock and the second class particle moves with speed $1-\lambda-\rho$. For large time $t$, we show that the position of the second class particle fluctuates on a $t^{1/3}$ scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time $t$.

Abstract: We consider a discrete model for anisotropic (2+1)-dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space-time limit to the (2+1)-dimensional additive stochastic heat equation (or Edwards-Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE.

Abstract: We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.

Abstract: Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the large time behavior for the covariance of the height gradients.