Inside singularity sets of random Gibbs measures

具有 Hartree 非线性的三维波动方程的不变 Gibbs 测度 I：测度

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures

本文构造并研究了具有 Hartree 非线性的三维波动方程的 Gibbs 测度，基于 $Φ^4_3$-model 方法处理非局部性，并确定了测度奇异性的精确阈值，为后续证明动力学不变性奠定基础。

具有 Hartree 非线性的三维波动方程的不变 Gibbs 测度 II：动力学

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics

本文证明了具有 Hartree 非线性的三维波动方程的 Gibbs 测度不变性，基于 para-controlled 方法建立局部理论，并提出新的全局化论证解决测度奇异性问题。

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic $Φ^4_3$-model. This result is the hyperbolic counterpart to seminal works on the parabolic $Φ^4_3$-model by Hairer '14 and Hairer-Matetski '18. The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled Ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates. The singularity …

Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments $\{ω(t)-ω(s)\}_{s<t}$ of $d$-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form $H(t-s,ω(t)-ω(s))$ that are invariant under uniform translations of paths. In such interactions we allow {\it{long-range}} dependence in the time variable (including power law decay up to $t\mapsto (1+t)^{-(2+\varepsilon)}$ for $\varepsilon>0$) and unbounded (singular) interactions (including singularities of the form $x\mapsto 1/|x|^p$ in $d\geq 3$ or $x\mapsto δ_0(x)$ in $d=1$) attached to the space…

On the Purity of the free boundary condition Potts measure on random trees

We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds are optimal for the Ising model and appear to be close to what we conjecture to be the true values up to a factor of 0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an iteration …