Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times

Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case

We prove scaling limit results for the finite-volume version of the inventory accumulation model of Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. In particular, we prove that the random walk associated with the finite-volume version of this model converges in the scaling limit to a correlated Brownian motion $\dot Z$ conditioned to stay in the first quadrant for two units of time and satisfy $\dot Z(2) = 0$. We also show that the times which describe complementary connected components of FK loops i…

Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent

We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various \emph{local estimates} for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular …

Ising (Conformal) Fields and Cluster Area Measures

We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.

Inventory Accumulation with $k$ Products

Sheffield (2011) proposed an inventory accumulation model with two types of products to study the critical Fortuin-Kasteleyn model on a random planar map, and showed that a two-dimensional inventory accumulation trajectory in the discrete model scales to a correlated planar Brownian motion. In this work, we generalize the inventory model to $k$ types of products for any integer $k \ge 2$, and prove that the corresponding trajectory scales to a $k$-dimensional Brownian motion with an appropriate covariance matrix.

Sandpiles and unicycles on random planar maps

We consider the abelian sandpile model and the uniform spanning unicycle on random planar maps. We show that the sandpile density converges to 5/2 as the maps get large. For the spanning unicycle, we show that the length and area of the cycle converges to the hitting time and location of a simple random walk in the first quadrant. The calculations use the "hamburger-cheeseburger" construction of Fortuin--Kasteleyn random cluster configurations on random planar maps.

Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $κ> 8$

We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin--Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield's hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including…

New critical exponent inequalities for percolation and the random cluster model

We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: The critical exponent inequalities $γ\leq δ-1$ and $Δ\leq γ+1$ hold for percolation and the random cluster m…

平面上的模糊Potts模型：标度极限与臂指数

The fuzzy Potts model in the plane: Scaling limits and arm exponents

该研究针对平面临界FK渗流染色得到的模糊Potts模型，证明其在共形不变标度极限假设下收敛于共形环系综变体，且离散与连续模型臂指数一致，进而确定了该模型的臂指数。

Basic properties of the infinite critical-FK random map

We investigate the critical Fortuin-Kasteleyn (cFK) random map model. For each $q\in [0,\infty]$ and integer $n\geq 1$, this model chooses a planar map of $n$ edges with a probability proportional to the partition function of critical $q$-Potts model on that map. Sheffield introduced the hamburger-cheeseburger bijection which maps the cFK random maps to a family of random words, and remarked that one can construct infinite cFK random maps using this bijection. We make this idea precise by a detailed proof of the local convergence. When $q=1$, this provides an alternative construction of the UI…

Critical Ising on the square lattice mixes in polynomial time

The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $\Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical g…

相位边界自旋系统的近似FKG不等式，及其在指数随机图模型中心极限定理中的应用

Approximate FKG inequalities for phase-bound spin systems, with applications to central limit theorems for exponential random graphs

本文研究了相位边界自旋系统中近似的FKG不等式，证明了在相变共存区域中，每个相内部确实满足近似的FKG不等式，并利用此结果完成了各个相内的中心极限定理证明，回答了Bianchi等人提出的问题。