48 problems

Half-plane self-avoiding-walk measure conjecture. The measure satisfies

Consider a self-avoiding walk of length $n$ on the lattice, and let its endpoint displacement from the origin be measured by the mean distance. Flory's conjecture. The mean displac…

Consider the extremal investor model with $x_0=0$, parameter $\alpha=1$, and Gaussian innovations of standard deviation $1$. Define … Thus $w_n$ is the greatest distance between…

Let $(x_n)_{n\geq 0}$ be the planar convex-hull-avoiding walk, and let $w_n$ be the distance of the farthest point on the path from the line $ox_n$, where $o$ is the origin. Three-…

Consider self-avoiding walk on ${\mathbb{Z}}^2$, and let its endpoint after $n$ steps be denoted by $S_n$. The endpoint scaling conjecture. The endpoint runs on scale … Determining…

Exponent-relation conjecture. The relationship between the exponents is generally

Optimal-exponent conjecture. The optimal exponent is

Intermediate diameter exponent conjecture. The typical diameter of $\mathsf{WST}^{\beta_n}(K_n)$ has an exponent strictly between $1/2$ and $1/3$.

SOP scaling conjecture. The same large-$n$ scaling is valid for self-overlapping polygons (SOP), modulo possible logarithmic corrections. This extends a scaling form observed in se…

Let $Lambda_L=\{-L,\ldots,L\}^d$ with $1\leq d\leq 3$, let $eta=\eta^{\mathrm{white}}$, and consider the regime $lambda_0\leq\lambda\leq 1$, where $lambda_0=L^{-\frac{4-d}{2}}$. Le…

Let $Lambda_L=\{-L,\ldots,L\}^d$ with $1\leq d\leq 3$, and let the disorder be $eta=\eta^{\mathrm{white}}$. Write $xi$ for the transversal fluctuation exponent in dimension $d$. Di…

In the Brownian motion case, the Hurst parameter is $H=1/2$, and $R_T$ denotes the radius of gyration of the self-repellent path. The available bounds are … for large $T$. Brownian…

Let $B_t^H$ be a self-repellent fractional Brownian motion with Hurst parameter $H$ in dimension $d$, and let $R_T$ denote its radius of gyration, … Suppose that, up to a possible…

Consider a two-dimensional weakly self-avoiding star polymer with $N$ branches, polymer length parameter $T$, self-avoidance parameter $\beta$, and radius $R$. Daoud–Cotton star-po…

Let $R_{d,\beta,T}=(E^{Q_T}[B_T^2])^{1/2}$ be the radius of weakly self-avoiding Brownian motion in dimension $d$, with penalization parameter $\beta>0$. Universal scaling exponent…

A self-avoiding random walk in two dimensions is expected to have an end-to-end distance of order $T^{3/4}$ after $T$ steps. Self-avoiding walk scaling conjecture. The end-to-end d…

Skew Brownian LIS exponent conjecture. There exists a function $\ell:[-1,1]\times(0,1)\to[1/2,1)$ such that, with probability tending to $1$ as $n\to\infty$,

Discrete–continuum exponent conjecture. With probability tending to $1$ as $n\to\infty$,

Critical exponent conjecture. For all $p\in(0,1)$ there exists $d(p)\in[\alpha_*(p),\beta^*(p)]$ such that, with probability tending to $1$ as $n\to\infty$,

Physicists' variance conjecture. The variance should satisfy

Let $\zeta$ and $z$ denote the Family–Vicsek roughness and dynamic exponents, respectively, for random surface growth models in dimension two. These exponents are expected to satis…

Let $t_n$ be a uniformly sampled tree of size approximately $n$ in the square-torus uniform spanning-tree model, and let $w(t_n)$ be its Euclidean width. The UST width exponent con…

In dimension $d$, let $\alpha$ denote the growth exponent governing the characteristic linear size of a DLA cluster with $n$ particles, so that the relevant size scales as…

Let ${\sf TDLA}_0(n,[0,+\infty)^2)$ be the square DLA tree with $n$ vertices, rooted at the corner ${\bf c}$, and let ${\bf v}$ be a uniformly sampled vertex of the tree. Write…

Near-critical speed conjecture. As $\lambda\searrow1$,