We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t\mapsto e^{-\alpha t^2}$ with $0<\alpha <4\pi /e$, we show that no point configuration in ${\mathbb{R}}^n$ of density $\rho$ can have energy less than $(\rho +o(1\left)\right)(\pi /\alpha {)}^{n/2}$ as $n\to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho (\pi /\alpha {)}^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling–Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi /\alpha {)}^{n/2}$ is no longer asymptotically sharp when $\alpha >\pi e$. As a consequence of our results, we obtain bounds in ${\mathbb{R}}^n$ for the minimal energy under inverse power laws $t\mapsto 1/t^{n+s}$ with $s>0$, and these bounds are sharp to within a constant factor as $n\to \infty$ with $s$ fixed.

We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time $\tau$) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the ${\tau }^{1/3}$-scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.

Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.

We investigate the stability of boundary layer solutions of the $2$-dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: $u^{\nu }(t,x,y)=\left(U^E\right(t,y)+U^{\mathrm{BL}}(t,\frac{y}{\sqrt{\nu }}),0)$, $0<\nu \ll 1.$ We show that if $U^{\mathrm{BL}}$ is monotonic and concave in $Y=y/\sqrt{\nu }$, then $u^{\nu }$ is stable over some time interval $(0,T)$, $T$ independent of $\nu$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{\mathrm{BL}}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr–Sommerfeld operator.