Physics considerations suggest that a theory of Liouville quantum gravity (LQG) should exist for all values of matter central charge $\mathbf{c} \in (-\infty,25)$. Probabilists have rigorously defined LQG as a random metric measure space for $\mathbf{c} < 1$; however, they have only very recently begun to explore LQG in the $\mathbf{c} \in (1,25)$ phase. We define a random metric associated to LQG for all $\mathbf{c} < 25$ by a collection of axioms analogous to axioms stated in the $\mathbf{c} < 1$ setting. We show that such a metric exists for each $\mathbf{c}$ by considering an approximating metric known as Liouville first passage percolation. Ding and Gwynne proved that these approximating metrics are tight in a suitably chosen topology; we show that every subsequential limit satisfies our axioms. In particular, our result defines a metric associated to LQG in the critical case $\mathbf{c} =1$.
The metrics we define for $\mathbf{c} \in (1,25)$ exhibit geometric behavior that sharply contrasts with the $\mathbf{c} < 1$ regime. We show that, for $\mathbf{c} \in (1,25)$, the metrics do not induce the Euclidean topology since they a.s. have a dense (measure zero) set of singular points, points at infinite distance from all other points. We use this fact to prove that a.s. the metric ball is not compact and its boundary has infinite Hausdorff dimension. Despite these differences, we demonstrate that many properties of LQG metrics for $\mathbf{c} < 1$ extend in some form to the entire range $(-\infty, 25)$. We show that the metrics are a.s. reverse Hölder continuous with respect to the Euclidean metric, are a.s. complete and geodesic away from the set of singular points, and satisfy the bounds for set-to-set distances that hold in the $\mathbf{c} < 1$ phase. Finally, we prove that the metrics satisfy a version of the (geometric) Knizhnik-Polyakov-Zamolodchikov (KPZ) formula.
We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity $c$ on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity $c$ corresponds to weighting the law of the surface by the $(-c/2)$-th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a L…
We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ …
There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge ${\mathbf{c}}_{\mathrm M}\in(-\infty,1]$. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating $γ$ times a variant of the planar Gaussian free field (GFF), where $γ\in(0,2]$ satisfies $\mathbf c_{\mathrm M}=25-6(2/γ+γ/2)^2$. Physics considerations suggest that LQG should also make sense in the regime when $\mathbf c_{\mathrm M}>1$. However, the behavior in this regime is rather mysterious in part because …
We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a h…
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学，法国国家科学研究中心，奥赛数学实验室）；University of Helsinki, Department of Mathematics and Statistics（赫尔辛基大学数学与统计系）；Aix Marseille Univ, CNRS, Centrale Marseille, I2M（艾克斯-马赛大学，法国国家科学研究中心，马赛中央理工学院，I2M）；Institut Universitaire de France (IUF)（法国高等研究院）
We consider a class of subcritical superprocesses $(X_t)_{t\geq 0}$ with general spatial motions and general branching mechanisms. We study the asymptotic behaviors of $\mathbf Q_{t,r}$, the distribution of $X_t$ conditioned on $X_{t+r}$ not being a null measure. We first give the existence of $\lim_{t\to \infty}\mathbf Q_{t,r}$ and $\lim_{r\to \infty}\mathbf Q_{t,r}$, and then show that an $L\log L$-type condition is equivalent to the existence of the double limits: $\lim_{r\to \infty} \lim_{t\to\infty}\mathbf Q_{t, r}$ and $\lim_{t\to \infty} \lim_{r\to\infty}\mathbf Q_{t, r}$. Finally, when…
Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space $(Ω,\mathcal{F},\mathbf{P})$. We show that if $\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each $w_n\in\mathbb{R}$, then \[{\mathbf{P}}(\limsup A_n)\geq\limsup_{n\to\infty} \frac{\displaystyle\big(\sum_{k=1}^n{w_k\mathbf{P}}(A_k)\big)^2}{\displaystyle\sum_{i=1}^n\sum_{j=1}^nw_iw_j{\mathbf{P}}(A_i\cap A_j)}.\]
Given a field $\{B(x)\}_{x\in\mathbf{Z}^d}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^d$, the generator of a suitable Markov process on $\mathbf{Z}^d,\,\,\mathcal{G},$ and sufficiently nice function $σ:[0,\infty)\to[0,\infty),$ we consider the influence of the parameter $λ$ on the behavior of the system, \begin{align*} \rm{d} u_t(x) = & (\mathcal{G}u_t)(x)\,\rm{d} t + λσ(u_t(x))\rm{d} B_t(x) \qquad[t>0,\ x\in\mathbf{Z}^d], &u_0(x)=c_0δ_0(x). \end{align*} We show that for any $λ>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\inft…
Brighton [Bri13] proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. $CD(0,\infty).$ In this paper, we provide a new proof of this result by using the reverse Poincaré inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the $CD(0,\infty)$ condition.
Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $\mathbf{g}\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic string theory \cite{Pol} (also called $2d$ bosonic string theory) and to Liouville quantum gravity. Then we show the convergence of Polyakov's partition function over the moduli space of Riemann surfaces in genus $\mathbf{g}\geq 2$ in the case of $D\leq 1$ boson. This is done by p…
Let $\{X(\mathbf{t}):\mathbf{t}=(t_1, t_2, \ldots, t_d)\in[0,\infty)^d\}$ be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function $r$ satisfying conditions $r(\mathbf{t})<1$ for every $\mathbf{t}\neq \mathbf{0}$ and $r(\mathbf{t})=1-\sum_{i=1}^d |t_i|^{α_i} + o(\sum_{i=1}^d |t_i|^{α_i})$, as $\mathbf{t}\to\mathbf{0}$, with constants $α_1, α_2, \ldots, α_d \in(0,2]$. The main result of this contribution is the description of the asymptotic behaviour of $P(\sup\{X(\mathbf{t}): \mathbf{t}\in\mathcal{J}^{\mathbf{x}}_{\mathbf{m}} \}…
Let $\{Y_{\mathfrak{B}}(x)\,:\,x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{γY_{\mathfrak{B}}(x)}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov--Hausdorff sense to a random metric on the…
In this paper we establish the limit of the empirical spectral distribution of quaternion sample covariance matrices. Suppose $\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}$ is a quaternion random matrix. For each $n$, the entries $\{x_{ij}^{(n)}\}$ are independent random quaternion variables with a common mean $μ$ and variance $σ^2>0$. It is shown that the empirical spectral distribution of the quaternion sample covariance matrix $\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*$ converges to the M-P law as $p\to\infty$, $n\to\infty$ and $p/n\to y\in(0,+\infty)$.
In this paper we provide a necessary and sufficient condition for the amenability of the automorphism group of Fraïssé structures and apply it to prove the non-amenability of the automorphism groups of the directed graph $\mathbf{S}(3)$ and the Boron tree structure $\mathbf{T}$. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $\mathrm{GL}(\mathbf{V}_\infty)$, where $\mathbf{V}_\infty$ is the countably infinite dimensional vector space over a finite field $F_q$, we show that the unique invariant measure on the univer…
This paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form $\mathbf S_n=\frac1n\mathbf B_n\mathbf X_n\mathbf X_n^*\mathbf B_n^*$, where $\mathbf B_n$ is a $p\times m$ non-random matrix and $\mathbf X_n$ is an $m\times n$ matrix consisting of i.i.d standard complex entries. $p/n\to c\in (0,\infty)$ as $n\to \infty$ while $m$ can be arbitrary. We proved that under some mild assumptions, with probability 1, there will be no eigenvalues in any closed interval contained in an open interval outside t…
Given a finite, connected graph $G$, the lamplighter chain on $G$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $G^\diamond={\mathbf Z}_2 \wr G$. The mixing time of the lamplighter chain on the torus ${\mathbf Z}_n^d$ is known to have a cutoff at a time asymptotic to the cover time of ${\mathbf Z}_n^d$ if $d=2$, and to half the cover time if $d \ge 3$. We show that the mixing time of the lamplighter chain on $G_n(a)={\mathbf Z}_n^2 \times {\mathbf Z}_{a \log n}$ has a cutoff at $ψ(a)$ times the cover time of $G_n(a)$ as $n \to \infty$, where $ψ$ is an explicit weakly…
We consider a random conductance model on the $d$-dimensional lattice, $d\in[2,\infty)\cap\mathbb{N}$, where the conductances take values in $(0,\infty)$ and are however not assumed to be bounded from above and below. We assume that the law of the random conductances is stationary and ergodic with respect to translations on $\mathbb{Z}^d$ and invariant with respect to reflections on $\mathbb{Z}^d$ and satisfies a similar moment bound as that by Andres, Deuschel, and Slowik (2015), under which a quenched FCLT holds. We prove a first-order Liouville theorem. In the proof we construct the subline…
