Typical ground states for large sets of interactions

典型余循环的Lyapunov指数熵谱

Entropy spectrum of Lyapunov exponents for typical cocycles

本文针对典型余循环，建立了Lyapunov指数水平集的拓扑熵与广义奇异值函数的拓扑压力之间的变分关系，刻画了所有Lyapunov指数水平集的大小。

Heat profile, level sets and hot spots of Laplace eigenfunctions

We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds for ground state Dirichlet eigenfunctions and look at the corresponding Neumann problem, namely the comparison of maximum temperatures on the interior and the boundary, the latter being partially motivated by the hot spots problem. We also investigate the proximity/distance of level sets of ground state Dirichlet eigenfunctions, some with analogous statements for Neum…

Typical long time behaviour of ground state-transformed jump processes

We consider a class of Lévy-type processes derived via a Doob-transform from Lévy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise lower and upper envelopes for the almost sure long time behaviour of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

多部分$p$-自旋模型的基态能量——部分提升RDT视角

Ground state energies of multipartite $p$-spin models -- partially lifted RDT view

本文利用部分提升随机对偶理论，为多部分$p$-自旋模型推导出匹配的基态能量上下界，在球面情形给出最优值，并探讨其与伊辛模型的联系。