6 problems

Let $B$ and $V$ be weight matrices satisfying Assumption, with $V\neq\mathrm{Id}$, and let $(\rho_t^\beta)_{t\geq 0}$ be the token-distribution evolution on ${\mathbb S}^{d-1}$. Le…

Partial homogenization conjecture. If the corresponding solutions of are denoted by $(\bm{X}^{(k)}_i(t))_{i\in[N_k]}$ and $(\bm{w}^{(k)}_{i,j}(t))_{i,j\in[N_k]}$, then

Let the assumptions of Theorem hold. For $\epsilon>0$, let $(\bm{X}^{(\epsilon)}(t),\bm{w}^{(\epsilon)}(t))$ solve, and let $(\bm{X}^{(0)}(t))$ solve the limiting system with the s…

Let the assumptions of Theorem hold, with the dynamics of $(\bm{X}_i(t))_{i\in[N]}$ and $(\bm{w}_{i,j}(t))_{i,j\in[N]}$ governed by the SDEs with weight-noise intensity $\eta>0$. L…

Consider a conservative long-range interacting particle system at thermal equilibrium with a tagged particle whose mean-field evolution is trivial. Let the linearized mean-field op…

Let $E(N,\lambda v)$ and $E(N-1,\lambda v)$ denote the ground-state energies of the trapped Bose gas with interaction scaling $\lambda$, and let $T$ and $v$ satisfy appropriate condit…