Fractal geometry of Airy$_{2}$ processes coupled via the Airy sheet

In last passage percolation models lying in the Kardar–Parisi–Zhang universality class, maximizing paths that travel over distances of order $n$ accrue energy fluctuates on scale $n^{1/3}$; and these deviate from linear interpolation their endpoints $n^{2/3}$. These energies may be viewed via a coordinate system respects scalings. What emerges by doing so is indexed $x,y\in \mathbb{R}$ $s,t\in with $s<t$ unit quantities $W_{n}(x,s;y,t)$ specifying scaled path moves coordinates between $(x,s)$ $(y,t)$. The space-time Airy sheet is, after parabolic adjustment, putative distributional limit $W_{\infty }$ this as $n\to \infty $. has recently been constructed (Dauvergne, Ortmann Virág (2020)) such Brownian percolation. article, we initiate study fractal geometry sheet. We prove difference profile given $\mathbb{R}\to \mathbb{R}:z\to W_{\infty }(1,0;z,1)-W_{\infty}(-1,0;z,1)$ nondecreasing process constant random neighbourhood almost every $z\in \mathbb{R}$; exceptional set violate condition surely Hausdorff dimension one-half. Points violation correspond to special behaviour for paths, result investigating behaviour, making use two inputs recent studies LPP; namely, regularity profiles, estimates rarity pairs disjoint begin end close each other.