11 problems

Let $\beta>\beta_c$, $h\geq h_w(\beta)$, and $\rho>0$. For each $N$, let $\mu_{\beta,\lambda,h,\Lambda_N}$ be the Ising-model measure in the box $\Lambda_N$ with external-field par…

Let $f$ be a continuous centred non-degenerate stationary Gaussian field on either $\mathbb{R}^d$ or $\mathbb{Z}^d$. A persistence, or hard wall event, is an event of the form…

Entropic-repulsion conjecture. If $g$ is strictly below the fluctuations of Brownian motion, then the limiting process conditioned on the corresponding restriction involving…

Let $n=2$ or $n=3$, let $V_N$ denote the finite domain of the membrane model, let $\psi=(\psi_x)_{x\in V_N}$ be its field, let $\operatorname{E}_N$ denote expectation, and let…

Consider an effective ensemble of $k$ ordered Ising polymer paths whose interactions have exponential decay with rate parameter $\chi>0$, competing with entropic repulsion between…

Let $v>\gamma^{\circ}$ and let $\mathbbm{Q}^{U_v}=\lim_{a\to\infty}\mathbbm{W}(\mathord\cdot\mid\mathcal{E}_a^{U_v})$ be any measure supplied by the critical-speed framework. Let…

Let $\gamma^{\circ}$ and $\gamma^{\bullet}$ be the critical speeds with $1<\gamma^{\circ}<\gamma^{\bullet}$, and let $\mathcal{E}_a^{U_v}$ be the conditioning events associated wit…

Let $U\subseteq\mathbbm{R}\times[0,\infty)$ be open with $(0,0)\in U$, and let $\operatorname{cost}(P)$ be the path cost. Assume that there is a unique path $P$ from $(0,0)$) to…

Let $\mathbbm{Q}^v$ be a limiting conditioned Wiener measure arising from the general framework, and write $L_x(\infty)=\lim_{T\to\infty}L_x(T)$ for the local time at level $x$. Mo…

Let $v$ be a parameter in the general approximation framework, with stopping times $\tau_a^v\to\infty$ and events $\mathcal{E}_a^v$ such that the conditioned measures converge to…

Let $\mathcal{E}$ be the event that Brownian motion has local time bounded everywhere by $1$, and let $\kappa$ denote the exponent measuring the strength of entropic repulsion thro…