Short- and long-time path tightness of the continuum directed random polymer

We consider the point-to-point continuum directed random polymer ($\mathsf{CDRP}$) model that arises as a scaling limit from $1+1$ dimensional directed polymers in the intermediate disorder regime. We show that the annealed law of a point-to-point $\mathsf{CDRP}$ of length $t$ converges to the Brownian bridge under diffusive scaling when $t \downarrow 0$. In case that $t$ is large, we show that the transversal fluctuations of point-to-point $\mathsf{CDRP}$ are governed by the $2/3$ exponent. More precisely, as $t$ tends to infinity, we prove tightness of the annealed path measures of point-to-point $\mathsf{CDRP}$ of length $t$ upon scaling the length by $t$ and fluctuations of paths by $t^{2/3}$. The $2/3$ exponent is tight such that the one-point distribution of the rescaled paths converges to the geodesics of the directed landscape. This point-wise convergence can be enhanced to process-level modulo a conjecture. Our short and long-time tightness results also extend to point-to-line $\mathsf{CDRP}$. In the course of proving our main results, we establish quantitative versions of quenched modulus of continuity estimates for long-time $\mathsf{CDRP}$ which are of independent interest.