arXiv subjects
Publications and source records attributed to Zhenhao Cai.
We prove that on the metric graph of $\mathbb{Z}^d$ for $d\in \{3,4,5\}$, clusters of the critical loop soup (or equivalently, clusters of the critical Gaussian free field level-set) admit a scaling limit. To the best of our knowledge, this presents the first scaling limit result for a percolation model on a three-dimensional lattice.
We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $\alpha$-th power of that candidate's vote count in the preceding round. For $\alpha=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $\alpha=1$. It was known that when $\alpha=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $\alpha>1$, it drops to order $\textit{loglog n.}$ In contrast, for $\alpha<1$, $\mathcal{T}$ is typically at least $\exp(\Omega(n))$. We also prove a sharp phase transition in the identity of the winner when $\alpha>1$. For every positive integer $k$, if $2^{1/k}<\alpha<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates.
We present and study the Pool model in $\mathbb{R}^2$, a rotationally symmetric analogue of Multi-Particle Diffusion-Limited Aggregation (MDLA), in which particles ("droplets") perform continuous-time random walks and are absorbed upon entering a circular pool initially centered at the origin. Each absorbed particle increases the pool's mass, and the pool expands so that its area grows accordingly, yielding a natural mass-preserving dynamics. A central tool which is of independent interest is a version of Kurtz's theorem for this model, depicting the field of particles conditioned on the growth of the pool as an independent non-homogeneous Poisson point process.
For the Gaussian free field on the metric graph of $\mathbb{Z}^d$ ($d\ge 3$), we consider the heterochromatic two-arm probability, i.e., the probability that two points $v$ and $v'$ are contained in distinct clusters of opposite signs with diameters at least $N$. For all $d\ge 3$ except the critical dimension $d_c=6$, we prove that this probability is asymptotically proportional to $N^{-[(\frac{d}{2}+1)\land 4]}$. Furthermore, we prove that conditioned on this two-arm event, the volume growth of each involved cluster is comparable to that of a typical (unconditioned) cluster; precisely, each cluster has a volume of order $M^{(\frac{d}{2}+1)\land 4}$ within a box of size $M$.
We prove that for the Gaussian free field (GFF) on the metric graph of $\mathbb{Z}^d$ (for all $d\ge 3$ except the critical dimension $d_c=6$), with uniformly positive probability there exist two distinct sign clusters of diameter at least $cN$ within a box of size $N$ such that their graph distance is less than $N^{-[(d-2)\vee (2d-8)]}$. This phenomenon contrasts sharply with the two-dimensional case, where the distance between two macroscopic clusters is typically on the order of their diameters, following from the basic property of the scaling limit ``conformal loop ensembles'' $\mathrm{CLE}_4$ (Sheffield-Werner'2001). As a byproduct, we derive that the number of pivotal edges for the one-arm event (i.e., the sign cluster containing the origin has diameter at least $N$) is typically of order $N^{(\frac{d}{2}-1)\land 2}$. This immediately implies that for the incipient infinite cluster (IIC) of the metric graph GFF, the dimension of cut edges (i.e., edges whose removal disconnects the IIC) equals $(\frac{d}{2}-1)\land 2$. Translated in the language of critical loop soups (whose clusters, by the isomorphism theorem, have the same distribution as GFF sign clusters), this leads to the analogous estimates where the counterpart of a pivotal edge is a pivotal loop at scale $1$. This result hints at the new and possibly surprising idea that already in dimension $3$, microscopic loops (even those at scale $1$) play a crucial role in the construction of macroscopic loop clusters.
The question of understanding the scaling limit of metric graph critical loop soup clusters and its relation to loop soups in the continuum appears to be one of the subtle cases that reveal interesting new scenarios about scaling limits, with a mixture of macroscopic and microscopic randomness. In the present paper, we show that in three dimensions, scaling limits of the metric graph clusters are strictly larger than the clusters of the limiting continuum Brownian loop soup. We actually show that the upper box counting dimension of the latter clusters is strictly smaller than $5/2$, while that of the former is $5/2$.
We establish sharp asymptotic bounds for the critical intensity of the Finitary Random Interlacements (FRI) model in four and higher dimensions with general trajectory length distributions. Our proof reveals that the construction of near-critical FRI clusters in four and higher dimensions is essentially analogous to a Galton-Watson process, whose expected number of offspring corresponds to the capacity of a random walk killed at the given length.
We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs $\widetilde{\mathbb{Z}}^d$ ($d\ge 3$). Specifically, we study the probability of connecting two general sets located on opposite sides of an annulus with inner and outer radii both of order $N$, where additional constraints are imposed on the distance of each set to the annulus. We show that for all $d \ge 3$ except the critical dimension $d=6$, this probability is of the same order as $N^{(6-d)\land 0}$ (serving as a correction factor) times the product of the two probabilities of connecting each set to the closer boundary of this annulus. The analogue for $d=6$ is also derived, although the upper and lower bounds differ by a divergent factor of $N^{o(1)}$. Notably, it was conjectured by Basu and Sapozhnikov (2017) that quasi-multiplicativity without correction factor holds for Bernoulli percolation on $\mathbb{Z}^d$ when $3\le d<6$ and fails when $d>6$. In high dimensions (i.e., $d>6$), taking into account the similarity between the metric graph GFF and Bernoulli percolation (which was proposed by Werner (2016) and later partly confirmed by the authors (2024)), our result provides support to the conjecture that Bernoulli percolation exhibits quasi-multiplicativity with correction factor $N^{6-d}$. During our proof of quasi-multiplicativity, numerous regularity properties, which are interesting in their own right, were also established. A crucial application of quasi-multiplicativity is proving the existence of the incipient infinite cluster (IIC), which has been completed by Basu and Sapozhnikov (2017) for Bernoulli percolation for $3\le d<6$. Inspired by their work, in a companion paper, we also utilize quasi-multiplicativity to establish the IIC for metric graph GFFs, for all $d\ge 3$ except for the critical dimension $6$.
In this paper, we establish the existence and equivalence of four types of incipient infinite clusters (IICs) for the critical Gaussian free field (GFF) level-set and the critical loop soup on the metric graph $\widetilde{\mathbb{Z}}^d$ for all $d\ge 3$ except the critical dimension $d=6$. These IICs are defined as four limiting conditional probabilities, involving different conditionings and various ways of taking limits: (1) conditioned on $\{0 \leftrightarrow{} \partial B(N)\}$ at criticality (where $0$ is the origin of $\mathbb{Z}^d$, and $\partial B(N)$ is the boundary of the box $B(N)$ centered at $0$ with side length $2N$), and letting $N\to \infty$; (2) conditioned on $\{0\leftrightarrow{} \infty\}$ at super-criticality, and letting the parameter tend to the critical threshold; (3) conditioned on $\{0 \leftrightarrow{} x\}$ at criticality (where $x\in \mathbb{Z}^d$ is a lattice point), and letting $x\to \infty$; (4) conditioned on the event that the capacity of the critical cluster containing $0$ exceeds $T$, and letting $T\to \infty$. Our proof employs a robust framework of Basu and Sapozhinikov (2017) for constructing IICs as in (1) and (2) for Bernoulli percolation in low dimensions (i.e., $3\le d\le 5$), where a key hypothesis on the quasi-multiplicativity is proved in our companion paper. We further show that conditioned on $\{0 \leftrightarrow{} \partial B(N)\}$, the volume of the critical cluster containing $0$ within $B(M)$ is typically of order $M^{(\frac{d}{2}+1)\land 4}$, as long as $N\gg M$. This phenomenon indicates that the critical cluster of the GFF or the loop soup exhibits self-similarity, which supports Werner's conjecture (2016) that such cluster has a scaling limit. Moreover, the exponent of $M^{(\frac{d}{2}+1)\land 4}$ matches the conjectured fractal dimension of the scaling limit proposed by Werner (2016).
We study the harmonic measure (i.e. the limit of the hitting distribution of a simple random walk starting from a distant point) on three canonical two-dimensional lattices: the square lattice $\mathbb{Z}^2$, the triangular lattice $\mathscr{T}$ and the hexagonal lattice $\mathscr{H}$. In particular, for the least positive value of the harmonic measure of any $n$-point set, denoted by $\mathscr{M}_n(\mathscr{G})$, we prove in this paper that $$[λ(\mathscr{G})]^{-n+c\sqrt{n}} \le \mathscr{M}_n(\mathscr{G})\le [λ(\mathscr{G})]^{-n+C\sqrt{n}},$$ where $λ(\mathbb{Z}^2)=(2+\sqrt{3})^2$, $λ(\mathscr{T})=3+2\sqrt{2}$ and $λ(\mathscr{H})=(\tfrac{3+\sqrt{5}}{2})^3$. Our results confirm a stronger version of the conjecture proposed by Calvert, Ganguly and Hammond (2023) which predicts the asymptotic of the exponent of $\mathscr{M}_n(\mathbb{Z}^2)$. Moreover, these estimates also significantly extend the findings in our previous paper with Kozma (2023) that $\mathscr{M}_n(\mathscr{G})$ decays exponentially for a large family of graphs $\mathscr{G}$ including $\mathscr{T}$, $\mathscr{H}$ and $\mathbb{Z}^d$ for all $d\ge 2$.
The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems (IPS). In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is the same as the kinetic Monte Carlo (KMC) method for the pairwise interacting particle system of size $N$. In the RBM-r algorithm, one randomly picks a small batch of size $p \ll N$, and only the particles in the picked batch interact among each other within the batch for a short time, where the weak interaction (of strength $\frac{1}{N-1}$) in the original system is replaced by a strong interaction (of strength $\frac{1}{p-1}$). Then one repeats this pick-interact process. This KMC algorithm dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step, and provides an unbiased approximation of the original force/velocity field of the interacting particle system. We give a rigorous proof of this approximation with an explicit convergence rate. In detail, we show that the Wasserstein-2 distance between first marginal distributions of IPS and RBM-r has an $O(\kappa^{1/4})$ upper bound, where $\kappa$ is the time step for choosing the random batch and the bound is independent of $N$. An improved $O(\kappa^{1/2})$ rate is also obtained when there is no diffusion in the system. Notably, the techniques in our analysis can potentially be applied to study KMC for other systems, including the stochastic Ising spin system.
For the critical level-set of the Gaussian free field on the metric graph of $\mathbb Z^d$, we consider the one-arm probability $θ_d(N)$, i.e., the probability that the boundary of a box of side length $2N$ is connected to the center. We prove that $θ_d(N)$ is $O(N^{-\frac{d}{2}+1})$ for $3\le d\le 5$, and is $N^{-2+o(1)}$ for $d=6$. Our upper bounds match the lower bounds in a previous work by Ding and Wirth up to a constant factor for $3\le d\le 5$, and match the exponent therein for $d=6$. Combined with our previous result that $θ_d(N) \asymp N^{-2}$ for $d>6$, this seems to present the first percolation model whose one-arm probabilities are essentially completely understood in all dimensions. In particular, these results fully confirm Werner's conjectures (2021) on the one-arm exponents: \begin{equation*} \text{(1) for}\ 3\le d d_c,\ θ_d(N)=N^{-2+o(1)}. \end{equation*} Prior to our work, Drewitz, Prévost and Rodriguez obtained upper bounds for $d\in \{3, 4\}$, which are very sharp although lose some diverging factors. In the same work, they conjectured that $θ_{d_c}(N) = N^{-2+o(1)}$, which is now established. In addition, in a recent concurrent work, Drewitz, Prévost and Rodriguez independently obtained the up-to-constant upper bound for $d=3$.
We prove that for $\mathbb{Z}^d$ ($d\ge 2$), the vertex-removal stability of harmonic measures (i.e. it is feasible to remove some vertex while changing the harmonic measure by a bounded factor) holds if and only if $d=2$. The proof mainly relies on geometric arguments, with a surprising use of the discrete Klein bottle. Moreover, a direct application of this stability verifies a conjecture of Calvert, Ganguly and Hammond [9] for the exponential decay of the least positive value of harmonic measures on $\mathbb{Z}^2$. Furthermore, the analogue of this conjecture for $\mathbb{Z}^d$ with $d\ge 3$ is also proved in this paper, despite vertex-removal stability no longer holding.
In this paper, we study the critical level-set of Gaussian free field (GFF) on the metric graph $\widetilde{\mathbb{Z}}^d,d>6$. We prove that the one-arm probability (i.e. the probability of the event that the origin is connected to the boundary of the box $B(N)$) is proportional to $N^{-2}$, where $B(N)$ is centered at the origin and has side length $2\lfloor N \rfloor$. Our proof is hugely inspired by Kozma and Nachmias [29] which proves the analogous result of the critical bond percolation for $d\geq 11$, and by Werner [51] which conjectures the similarity between the GFF level-set and the bond percolation in general and proves this connection for various geometric aspects.
In this paper, we prove the exact orders of critical intensity $u_*(T)$ in Finitary Random Interlacements (FRI) in $\mathbb{Z}^d, \ d\ge 3$ with respect to the expected fiber length $T$. We show that as $T\to\infty$, $u_*(T)\sim T^{-1}, \ d\ge 5;$ $u_*(T)\sim T^{-1}\log T, \ d=4;$ $u_*(T)\sim T^{-1/2},\ d=3.$ Our estimates also give the order of magnitude at which the percolative phase transition with respect to $T$ takes place.
We prove that the critical percolation parameter for Finitary Random Interlacements (FRI) is continuous with respect to the path length parameter $T$. The proof uses a result which is interesting on its own right; equality of natural critical parameters for FRI percolation phase transition.
In this paper, we show several rigorous results on the phase transition of Finitary Random Interlacement (FRI). For the high intensity regime, we show the existence of a critical fiber length, and give the exact asymptotic of it as intensity goes to infinity. At the same time, our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity.
In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, we show that for $d=3,4$, FRI is NOT stochastically monotone as fiber length increasing. At the same time, numerical evidences still strongly support the existence of a unique and sharp phase transition on the existence of a unique infinite cluster, while the critical value for phase transition is estimated to be an inversely proportional function with respect to the system intensity.