Related papers: Invariant Gibbs measures for the three dimensional…
We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov…
This paper concerns Gibbs measures $\nu$ for some nonlinear PDE over the $D$-torus ${\bf T}^D$. The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2 - \int_{{\bf T}^D} \vert u\vert^p$ has canonical equations with solutions in…
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which…
Mathematical models in equilibrium statistical mechanics describe physical systems with many particles interacting with an external force and with one another. Gibbs measure is a fundamental concept in this theory. In existing literature…
We give an introductory account of the recently identified gauge invariance of the equilibrium statistical mechanics of classical many-body systems [J. M\"uller et al., Phys. Rev. Lett. Phys. Rev. Lett. 133, 217101 (2024)]. The gauge…
We consider discrete analogues of two well-known open problems regarding invariant measures for dispersive PDE, namely, the invariance of the Gibbs measure for the continuum (classical) Heisenberg model and the invariance of white noise…
In this paper we prove the existence of an invariant measure for the cubic NLS $$i\partial_t u + \bigtriangleup u - |u|^2 u = 0$$ on the real line in the sense that we prove the existence of a measure $\rho$ supported by non-localised…
We prove the local wellposedness of the (renormalized) parabolic $\Phi^4_3$ model associated with the harmonic oscillator on $\mathbb{R}^3$, that is, the equation formally written as \begin{equation*} \partial_t X + HX= -X^3+\infty\cdot X +…
We show that the Kuperberg invariant of the Weeks manifold with any framing is a gauge invariant of finite-dimensional Hopf algebras, which provides the first example of gauge invariants of general finite-dimensional Hopf algebras via…
In this paper, we show that the Gibbs measure of the stochastic hyperbolic sine-Gordon equation on the circle is the unique invariant measure for the Markov process. Moreover, the Markov transition probabilities converge exponentially fast…
The Higgs mechanism gives mass to Yang-Mills gauge bosons. According to the conventional wisdom, this happens through the spontaneous breaking of gauge symmetry. Yet, gauge symmetries merely reflect a redundancy in the state description and…
We consider $k$-equivariant wave maps from the exterior spatial domain $\mathbb{R}^3\backslash B(0,1)$ into the target $\mathbb{S}^3$. This model has infinitely many topological solitons $Q_{n,k}$, which are indexed by their topological…
We consider a gas whose each particle is characterised by a pair $(x,v_x)$ with the position $x\in \mathbb R^d$ and the velocity $v_x\in \mathbb R^d_0= \mathbb R^d\setminus \{0\}$. We define Gibbs measures on the cone of vector-valued…
The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural…
We present a moduli space for all hyperbolic basic sets of diffeomorphisms on surfaces that have an invariant measure that is absolutely continuous with respect to Hausdorff measure. To do this we introduce two new invariants: the measure…
We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic…
We study the hyperbolic sine-Gordon model, with a parameter $\be^2 > 0$, and its associated Gibbs dynamics on the two-dimensional torus. By introducing a physical space approach to the Fourier restriction norm method and establishing…
We consider irreversible translation-invariant interacting particle systems on the $d$-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy…
In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $\rho (X) =1$. And we show that all walls are geodesic in the normalized space with respect to…
We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of…