7 problems

Let Assumption $\mathrm{(function)}$ hold, and set $q=q(N)=e^{-\gamma(N)/N}$, where $\gamma(N)\to+\infty$ and $\gamma(N)\ll N$. Let $\lambda^r_j(N)$ denote the parts of the random…

Let $\alpha$ and $\beta$ be parameters satisfying … Let $t$ be the interface parameter, and let a smooth region mean a smooth region of the limit shape to the left of the interface…

Let $\mathsf{Ht}_\mathcal N$ be the random height function of the model at scale $\mathcal N$, let $\mathbb E[\mathsf{Ht}_\mathcal N]$ be its expectation, let $\pounds$ denote the…

Circular limit-shape conjecture. As $n\to\infty$, the limit-shape curve is

Consider the cubiculated box … Let $\mathbf T,\mathbf T_0,\mathbf T_1$ be independent uniformly distributed random tilings of $\mathcal R$, and let $\operatorname{Tw}$ denote the t…

Fix an arbitrary tilable polygonal domain $\Upsilon$ on the triangular grid. Let $H_L(x,y)$ be the random height function of a uniformly random lozenge tiling of the dilated domain…

Let $(\mathpzc{X}(t))_{t=0,1,\dots}$ be the particle process arising from the shuffling algorithm for random tilings of the Aztec diamond, and let $(\mathbf{X}(t))_{t\geq 0}$ denot…