27 problems

The delayed Poissonian mean-field model is an interacting-particle system with Poisson-like attributes, and its corresponding mean-field PDE is intended to describe the large-parti…

Domain-of-attraction conjecture. An analogous domain-of-attraction statement should remain valid for broader mean-field graph sequences, at least when the dual random walk mixes on…

Let ${\vec m}$ be the parameter vector in the definition of the functional ${\rm P}_k$ in (Pk). Convexity conjecture. The functional ${\rm P}_k$ is convex in ${\vec m}$. The conjec…

Generalized Parisi conjecture. Replica symmetry breaking, when properly applied, should always yield the correct free energy; in particular,

Single-site approximation conjecture. There is a value $\lambda(\beta,\varepsilon,h_0)$ such that

Boltzmann-state $x$-representability conjecture. The Boltzmann state of mean field spin glass models is $x$-representable.

Let a network structure and a lumping partition be such that the relevant combinatorial calculations can be expressed in terms of the hypergeometric distribution. The lumped transi…

Uniqueness conjecture. Equation (11) has a unique solution.

Mean-field busy-probability comparison. For $<$,

Critical-point uniqueness conjecture. The measure $sigma$ is the unique critical point of the free-energy functional $\mathcal F_K$ in this range.

Let $X$ be the solution of the integrate-and-fire system on a partition ${\boldsymbol{\mathcal{P}}}^I$, and let $\mathbf{G}$ be the transition kernel of the associated Markov chain…

Let $m(\varepsilon)$ denote the probability vector in the preceding first-order expansion, and let $\mu^{\text{guess}}_\varepsilon$ be a measure on the angular variable $\theta$. L…

Possible-number-of-solutions conjecture. For any finite $k$, there exist $cbeta$, $h$, and $cP$ such that the equation above has any number of solutions in the set…

Let the network SIR model be the deterministic mean-field model associated with a stochastic SIR process on a contact network, and let the network's average degree be the mean numb…

Higher-order interaction generalization conjecture. An appropriate generalization of these exponential bounds should hold for stochastic population processes with higher-order inte…

Global convergence conjecture. If $\rho_0$ is a smooth distribution with full support, then the dynamics described by this flow should converge and the population risk…

Let $f_t^{(n)}$ be the empirical distribution of the population in the one-dimensional Attraction-Repulsion Model, and let $\tau$ be the interaction threshold. For parameters…

Consider the isolated neuron in the case $J=0$, with rate function … and drift … An invariant distribution is a probability distribution preserved by the linear neuron dynamics, an…

Countable-label extension conjecture. We conjecture that the results of the paper continue to hold for this countable-label model.

The mean-field forest fire model evolves by burning the edges of a component when a vertex clock rings; its limiting dynamics are described by coupled forest fire equations, with a…

Let $X_i$ be the payment of bank $i$, let $A_i^{\mathrm{in}}$ be its incoming-neighbor set, and let $A_j$ be the outgoing-neighbor set of bank $j$. Let $Z$ be a common shock, $Z_i$…

Correlated-shock fixed-point conjecture. As $n\to\infty$,

Let $X_i$ denote the payment of bank $i$, with constants $k$, $a$, and $y$, and let $Z_i$ be independent shocks. Let $p$ denote the default probability and set $m=k+a+E(X_i)$. Asym…

Let $g^+$ and $g^-$ denote the positive and negative density functions in the space-homogeneous deterministic mean field Cross model. The steady states are classified into types…

Let $\mathcal X_g$ be the set of symmetric $n\times n$ matrices with zero diagonal defined by the conditions that, for some $\lambda\in\mathbb R$,…