36 problems

Let the once-reinforced random walk (ORRW) with reinforcement parameter $c>0$ be run on the integer lattice $\mathbb{Z}^d$. Sidoravicius's recurrence conjecture. The walk is recurr…

Consider a stronger-than-linear edge-reinforced random walk whose transition probabilities are proportional to a positive function $f$ of the number of edge crossings, under the hy…

Let $N$ and $F$ be the nest and food source, respectively, in a finite graph, and let ants make successive random walks from $N$ that stop when they hit $F$, updating pheromone lev…

Let $W$ be once-reinforced random walk on $\mathbb{Z}^d$ with fixed reinforcement parameter $a>0$, and let $t$ denote time. The conjecture concerns the regime in which $a$ is fixed…

Let $\Sigma$ be the simplex of limiting edge-traversal proportions, let $V$ be its set of vertices, and let $\mathcal{D}_0=\{\Theta\in\Sigma\setminus\partial\Sigma\}$,…

Let $G=(V,E)$ be a connected locally finite graph, and consider a self vertex-reinforced random walk on $G$. The walk may eventually become confined to a random fixed subset of $V$…

Let a strongly edge-reinforced random walk evolve on an arbitrary graph with bounded degree, with reinforcement increasing sufficiently fast for the strongly reinforced regime. Sel…

Let ${\mathbf{X}}=(X_t)_{t\in\mathbb{R}_{+}}$ be a strongly vertex-reinforced jump process ${\rm VRJP}(\ell,w)$ on a graph of bounded degree, with initial local times bounded below…

Consider the two-walker model in which the moving walker is selected randomly, and let $M^{\rm rd}_\infty\in[0,1]$ be the limit of the left edge weight proportion. Let…

Consider the two-walker model in which the walker to move is selected randomly, and let $M^{\rm rd}_\infty$ denote the limiting proportion of the left edge weight. The alternating-…

Biased edge-reinforced random walk transience conjecture. The $\lambda^\ast$-biased edge-reinforced random walk is transient whenever $\lambda\neq1$.

Alternating two-player urn distribution conjecture. The random variable $Y\in[0,1]$ has a density with respect to Lebesgue measure on $[0,1]$, is not Beta-distributed, and has a di…

Let $\alpha_c^1(\delta)$ be the critical exponent for exponential integrability of the cover time of a $\delta$-once-reinforced random walk, and let $X$ start from a specified vert…

Consider a finite graph with subgraph state space $V_S$, rate function $\Lambda_\delta$, and invariant probability $\mu_0$ of the transition kernel $\mathbf{p}$. Let $\mathfrak{b}$…

Let $G=(V,E)$ be a finite connected graph, let $uinmathscr{P}(V)$, and let $Idelta$ denote the good rate function for the empirical measure process of the $delta$-once-reinforced…

Let $M=(\Omega,P)$ be a transient Markov chain, and let $X=(X_n)_{n\geq 0}$ be a trajectory of $M$. The partially exchangeable $\sigma$-algebra of $X$ is the exchangeable $\sigma$-…

Stable-configuration conjecture. If the reinforcement is strong enough, every linearly stable configuration is a union of trees of diameter at most $3$.

Sidoravicius–Beffara–Keane conjecture. There exists a phase transition in the behavior of the ORRW as the reinforcement parameter $$ varies.

Davis's conjecture. The ORRW is recurrent in dimension two.

Three-site localization conjecture. There exists a reinforcement weight function $w$ such that the corresponding VRRW on ${\mathbb Z}$ is recurrent but spends asymptotically all of…

Dichotomy conjecture. One has

Phase-transition conjecture. The ORRW on $\mathbb{Z}^d$ undergoes a phase transition between recurrence and transience as the reinforcement parameter varies.

Let $d$ be fixed, and consider the once-reinforced walk on a $d$-regular tree with forward-edge weights $u_0$ before reinforcement and $u_1$ after reinforcement. Monotonicity conje…

Let $\mathbb{Z}^d$ be the $d$-dimensional integer lattice, and let the linearly edge-reinforced random walk (ERRW) assign to each edge an initial weight $w$ plus the number of time…

Let $f$ be a reinforcement function, and call it strong if … Consider a reinforced random walk (RRW) on the triangle, where vertices are connected by its three edges and traversal…