6 problems

Let $X_t$ be the one-dimensional Brownian polymer process with interaction function $f$. Assume that the interaction is symmetrical and repulsive, with limited interaction, namely…

Let $\mu_{n,T}^{\mathfrak{f}}$ denote the line-ensemble measure with Lebesgue-free boundary conditions, and let $\mu_{\infty}^{\mathfrak{0}}$ be the limiting measure obtained for z…

The diffusion coefficient $D(t)$ describes the growth of the mean square displacement through $\mathbf E[|X(t)|^2]$ of order $tD(t)$. In the analogous self-interacting diffusions,…

Let $E(t)={\mathbf E}(X(t)^2)$, and suppose that for some $-1<\alpha<1$ the infrared bounds … hold. Diffusivity exponent conjecture. Under these conditions, the true asymptotic ord…

Tóth–Werner scaling conjecture. Under these assumptions, $X(t)/t^{2/3}$ converges in law. The conjecture predicts the $t^{2/3}$ fluctuation scale by analogy with discrete space-tim…

Durrett–Rogers law of large numbers conjecture. Under these assumptions, $X(t)/t\to0$ almost surely. This conjecture concerns the long-time displacement of a one-dimensional self-i…