arXiv subjects

Pengfei Tang

Publications and source records attributed to Pengfei Tang.

16 recordsLinked to original sources

arXiv2026

Negative correlation for the random-cluster model below one on high-degree regular graphs

The random-cluster model with cluster parameter $0<Q<1$ is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first show that restricting the problem to regular graphs of diverging degree does not essentially weaken the pairwise conjecture: validity for all such graph sequences, even when restricted to edge pairs at distance $o(d/\log d)$, is equivalent to validity on arbitrary finite graphs. We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed $p\in(0,1)$, all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance $o(d/\log d)$, uniformly in $Q\in[0,1)$ and in the choice of the two edges. In particular, the result includes the $Q=0$ endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.

math.PR↗

arXiv2026

From second moments to pairwise negative correlation: applications to minimal and uniform spanning trees

We uncover a close connection between the second moment of the degree of a typical vertex in a random subgraph and the pairwise negative correlation (p-NC) property. On one hand, we exploit this connection to prove the p-NC property for non-adjacent edges in minimal spanning trees on complete graphs. On the other hand, we apply the classical p-NC property of uniform spanning trees to derive a universal upper bound on the second moment of the degree of a uniformly chosen vertex in uniform spanning trees on finite, connected, regular graphs, thereby resolving an open question posed by Nachmias and Peres. Furthermore, we determine that the optimal upper bound is exactly 6, and the method for achieving this optimal bound is interesting in itself -- the proof uses Edmonds' matroid polytope theorem.

math.PR↗

arXiv2026

Pairwise Negative Correlation for Uniform Spanning Subgraphs of the Complete Graph

We investigate the pairwise negative correlation (p-NC) property for uniform probability measures on several families of spanning subgraphs of the complete graph $K_n$. Motivated by conjectured negative dependence properties of the random-cluster model with $q<1$, we focus on three natural families: the set of all connected spanning subgraphs, the set of forests with exactly $k$ components, and the set of connected spanning subgraphs with excess $k$, where $k$ is a fixed integer. We prove that for each of these families, the associated uniform measure satisfies the p-NC property provided $n$ is sufficiently large. Our results extend earlier work on uniform forests and provide the first verification of the p-NC property for uniform connected subgraphs and their truncations on complete graphs.

math.PR↗

arXiv2025

Maximum flow and self-avoiding walk on bunkbed graphs

We consider two combinatorial models on bunkbed graphs: maximum flow and self-avoiding walks. A bunkbed graph is defined as the Cartesian product $G\times K_2$, where $G$ is a finite graph and $K_2$ is the complete graph on two vertices, labelled $0$ and $1$. For the maximum flow problem, we show that if the bunkbed graph $G\times K_2$ has non-negative, reflection-symmetric edge capacities, then for any $u, v\in V(G)$, the maximum flow strength from $(u,0)$ to $(v,0)$ in $G\times K_2$ is at least as large as that from $(u,0)$ to $(v,1)$. For the self-avoiding walk model on a bunkbed graph $G\times K_2$, we investigate whether there are more self-avoiding walks from $(u,0)$ to $(v,1)$ than from $(u,0)$ to $(v,0)$. We prove that this holds when $G=K_n$ is a complete graph and $n$ is sufficiently large. Additionally, we provide examples where the statement does not holds and pose the question of whether it remains true when $\{u,v\}$ is not a cut-edge of $G$.

math.PR↗

arXiv2023

The union of independent USFs on $\mathbb{Z}^d$ is transient

We show that the union of two or more independent uniform spanning forests (USF) on $\mathbb{Z}^d$ with $d\geq 3$ almost surely forms a connected transient graph. In fact, this also holds when taking the union of a deterministic everywhere percolating set and an independent $\epsilon$-Bernoulli percolation on a single USF sample.

math.PR↗

arXiv2023

Evoke: Evoking Critical Thinking Abilities in LLMs via Reviewer-Author Prompt Editing

Large language models (LLMs) have made impressive progress in natural language processing. These models rely on proper human instructions (or prompts) to generate suitable responses. However, the potential of LLMs are not fully harnessed by commonly-used prompting methods: many human-in-the-loop algorithms employ ad-hoc procedures for prompt selection; while auto prompt generation approaches are essentially searching all possible prompts randomly and inefficiently. We propose Evoke, an automatic prompt refinement framework. In Evoke, there are two instances of a same LLM: one as a reviewer (LLM-Reviewer), it scores the current prompt; the other as an author (LLM-Author), it edits the prompt by considering the edit history and the reviewer's feedback. Such an author-reviewer feedback loop ensures that the prompt is refined in each iteration. We further aggregate a data selection approach to Evoke, where only the hard samples are exposed to the LLM. The hard samples are more important because the LLM can develop deeper understanding of the tasks out of them, while the model may already know how to solve the easier cases. Experimental results show that Evoke significantly outperforms existing methods. For instance, in the challenging task of logical fallacy detection, Evoke scores above 80, while all other baseline methods struggle to reach 20.

cs.CL↗

arXiv2023

DeepTagger: Knowledge Enhanced Named Entity Recognition for Web-Based Ads Queries

Named entity recognition (NER) is a crucial task for online advertisement. State-of-the-art solutions leverage pre-trained language models for this task. However, three major challenges remain unresolved: web queries differ from natural language, on which pre-trained models are trained; web queries are short and lack contextual information; and labeled data for NER is scarce. We propose DeepTagger, a knowledge-enhanced NER model for web-based ads queries. The proposed knowledge enhancement framework leverages both model-free and model-based approaches. For model-free enhancement, we collect unlabeled web queries to augment domain knowledge; and we collect web search results to enrich the information of ads queries. We further leverage effective prompting methods to automatically generate labels using large language models such as ChatGPT. Additionally, we adopt a model-based knowledge enhancement method based on adversarial data augmentation. We employ a three-stage training framework to train DeepTagger models. Empirical results in various NER tasks demonstrate the effectiveness of the proposed framework.

cs.IR↗

arXiv2022

A subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate

We construct an example of a subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate. This answers a question of Amir and Yang (2022) in the negative.

math.PR↗

arXiv2022

The wired minimal spanning forest on the Poisson-weighted infinite tree

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.

math.PR↗

arXiv2021

Constructing Sub-scale Surrogate Model for Proppant Settling in Inclined Fractures from Simulation Data with Multi-fidelity Neural Network

Particle settling in inclined channels is an important phenomenon that occurs during hydraulic fracturing of shale gas production. Generally, in order to accurately simulate the large-scale (field-scale) proppant transport process, constructing a fast and accurate sub-scale proppant settling model, or surrogate model, becomes a critical issue, since mapping between physical parameters and proppant settling velocity is complex. Previously, particle settling has usually been investigated via high-fidelity experiments and meso-scale numerical simulations, both of which are time-consuming. In this work, a new method is proposed and utilized, i.e., the multi-fidelity neural network (MFNN), to construct a settling surrogate model, which could utilize both high-fidelity and low-fidelity (thus, less expensive) data. The results demonstrate that constructing the settling surrogate with the MFNN can reduce the need for high-fidelity data and thus computational cost by 80%, while the accuracy lost is less than 5% compared to a high-fidelity surrogate. Moreover, the investigated particle settling surrogate is applied in macro-scale proppant transport simulation, which shows that the settling model is significant to proppant transport and yields accurate results. This opens novel pathways for rapidly predicting proppant settling velocity in reservoir applications.

physics.flu-dyn↗

arXiv2021

Return probabilities on nonunimodular transitive graphs

Consider simple random walk $(X_n)_{n\geq0}$ on a transitive graph with spectral radius $\rho$. Let $u_n=\mathbb{P}[X_n=X_0]$ be the $n$-step return probability and $f_n$ be the first return probability at time $n$. It is a folklore conjecture that on transient, transitive graphs $u_n/\rho^n$ is at most of the order $n^{-3/2}$. We prove this conjecture for graphs with a closed, transitive, amenable and nonunimodular subgroup of automorphisms. We also conjecture that for any transient, transitive graph $f_n$ and $u_n$ are of the same order and the ratio $f_n/u_n$ even tends to an explicit constant. We give some examples for which this conjecture holds. For a graph $G$ with a closed, transitive, nonunimodular subgroup of automorphisms, we prove a weaker asymptotic behavior regarding to this conjecture, i.e., there is a positive constant $c$ such that $f_n\geq \frac{u_n}{cn^c}$.

math.PR↗

arXiv2020

A note on some critical thresholds of Bernoulli percolation

Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons' conjecture of $p_{\mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahn's modified question. The key observation is that in Kahn's modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-Copin and Tassion (\textit{Comm. Math. Phys. Volume 343, 725-745.} (2016)). We also study some applications for Bernoulli percolation on periodic trees.

math.PR↗

arXiv2019

Weights of uniform spanning forests on nonunimodular transitive graphs

Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel--Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We also show that each tree of the free uniform spanning forest is heavy and has branching number bigger than one on a free product of a nonunimodular transitive graph with one edge when the free uniform spanning forest is not the same as the wired.

math.PR↗

arXiv2019

A reverse Aldous/Broder algorithm

The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here we show that the tree formed by all the last-exit edges also has the law of a uniform spanning tree. This answers a question of Tom Hayes and Cris Moore from 2010. The proof relies on a bijection that is related to the BEST theorem in graph theory. We also give other applications of our results, including new proofs of the reversibility of loop-erased random walk, of the Aldous--Broder algorithm itself, and of Wilson's algorithm.

math.PR↗

arXiv2018

Heavy Bernoulli-percolation clusters are indistinguishable

We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm (1999) in the Bernoulli percolation case and confirms a conjecture of Schonmann (2001). We also prove that every infinite cluster of Bernoulli percolation on a nonamenable quasi-transitive graph is transient almost surely.

math.PR↗

arXiv2016

Superlinearity of geodesic length in 2$D$ critical first-passage percolation

First-passage percolation is the study of the metric space $(\mathbb{Z}^d,T)$, where $T$ is a random metric defined as the weighted graph metric using random edge-weights $(t_e)_{e\in \mathcal{E}^d}$ assigned to the nearest-neighbor edges $\mathcal{E}^d$ of the $d$-dimensional cubic lattice. We study the so-called critical case in two dimensions, in which $\mathbb{P}(t_e=0)=p_c$, where $p_c$ is the threshold for two-dimensional bond percolation. In contrast to the standard case $( 1$ such that with probability at least $1-e^{-\|x\|_1^c}$, the minimal length geodesic from $0$ to $x$ has at least $\|x\|_1^s$ number of edges. Our proofs combine recent ideas to bound $T$ for general critical distributions, and modifications of techniques of Aizenman-Burchard to estimate the Hausdorff dimension of random curves.

math.PR↗