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I consider classical $\phi^4$-theory on the lattice. The model is defined in finite volume with Hamiltonian \begin{align*} H(\phi) = - \sum_{x \sim y} J_{x,y} \phi_x \phi_y \end{align*} and a-priori ...

Let $\Lambda_n = \{ -n,\ldots,n\}^2$ and let $\{X_i\}_{i\in \Lambda_n}$ be the family of $\mathbb{R}$-valued of random variables with the density proportional to $\exp(-\sum_{i\sim j} |X_i - X_j|),$ ...

Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...

Motivating example: Take the (wired) random cluster model $\phi^1_{p,q}$ with parameter $q$ (see http://arxiv.org/abs/1707.00520 for an introduction). It is now known on $\mathbb{Z}^2$ that it has a ...

Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as $$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...