25 problems

Consider a superimposition formed by combining the edges of two groves on different sets of initial conditions, allowing translations, so that the result is a multiset of edges. Th…

Let $mu=mun satisfy … and let$\boldsymbol{T}_n be the height-biased random tree of size $n$. Write $deg{boldsymbol{T}n}(varnothing)$ for the number of children of its root. **Root…

Consider the Fleming–Viot particle systems $(\mathbf X^r_t)_{t}$ with empirical measures $\pi(\mathbf X^r_t)$, in the regime where both $n_r$ and $\underline\lambda_r$ tend to infi…

Center-realization conjecture. An indecomposable subalgebra

Condensable-algebra fusion conjecture. Their relative tensor product satisfies

Separable-algebra decomposition conjecture. There is a separable algebra $B'\in\CM$ such that

Domain-wall decomposition conjecture. Any gapped domain wall of the form

Let $\textsc{NAE-SAT}$ be the constraint satisfaction model at constraint density $[...$

Standard-condition condensation conjecture. The condensation conclusion of Theorem 3.11 should remain valid if its condition

Let $S_{\star}$ be the set of sites on which the condensate may reside, and let $r(x,y)$ be the jump rates of the underlying random walk. Let $Y^{\mathrm{rv}}(\cdot)$ be the Markov…

Let $S$ be the finite state space of the underlying random walk, with jump rates $r(x,y)$, and let $S_{\star}$ be the set of sites on which the condensate may reside. Define the Ma…

Domination conjecture. The first random model should be dominated by the second random model in the same sense: the second random model should have higher moments of every order an…

Let $m(t)$ denote the time-dependent memory parameter in Regime $R3$, let $\mu_\epsilon$ be the mean of the fitness increments, and let the cumulative fitness distribution be the d…

Let $m\in\mathbb{N}$, let $G_t$ be a preferential attachment graph in Regime $R2$, and let $BB(m)$ denote the corresponding benchmark model. Condensation-transfer conjecture in…

Let $cal F$ be an arbitrary bosonic $(D+1)$-dimensional topological order. For each $dleq D$, let $cal F[S^d]$ denote its compactification along the $d$-sphere. Compactification co…

Let $H_{kin}$ be the kinetic, or hopping, part of the Bose–Hubbard Hamiltonian, let $u$ be the parameter of the quantum infinite-temperature state … where $A$ is an observable and…

Let $\Xi_t(f,g)$ denote the mass distribution of families with fitness in the interval determined by $f$ and $g$, and let $\alpha$, $\beta$, and $\gamma$ be the model parameters. U…

Critical-parameter conjecture. The behavior of the functions $F_b(\cdot)$ changes at $b=3/2$: $F_b(\cdot)$ is positive for $b\in(1,3/2]$, so that the process is attractive, whereas…

Probabilistic convergence conjecture. This convergence holds not only in expectation, but also in probability. Establishing this convergence in probability is the subject of an ong…

Consider models on a bounded and continuous state space in which effects similar to mutation and selection compete effectively, and suppose that a wave moving towards the maximal s…

In Kingman's model, the wave moving towards genes of maximal fitness is described by a gamma distribution. Universality conjecture. This wave shape is a universal phenomenon that c…

Blow-up–condensation time conjecture. The physical-literature conjecture is

Let $Y_{(1)}$ be the largest box size in the random allocation, and suppose $0<\nu<\lambda$. Nonvanishing maximum-size subsequence conjecture. There exists $\varepsilon>0$ and a su…

Let $\mathbf{w}=(w_k)_{k\geqslant0}$ be a weight sequence with radius parameter $\rho>0$. Let $Y_{(1)}$ denote the largest box size in the associated random allocation, with…

Let a homogeneous random potential contain an interacting Bose gas, and consider generalized Bose–Einstein condensation in this random medium. The Kac–Luttinger conjecture. Condens…