arXiv subjects

Chuwen Wang

Publications and source records attributed to Chuwen Wang.

9 recordsLinked to original sources

arXiv2026

A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra

We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in $\mathbb{H}^3$. By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number $L_f(P)$, determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula $\mathbb{E}[L_f(P)] = 2π- (π/3)\mathbb{E}[deg(f)]$. Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular random trivalent hyperbolic polyhedron is parabolic precisely when $\mathbb{E}[L_f(P)] = 0$, and hyperbolic when $\mathbb{E}[L_f(P)] < 0$. Thus, global conformal type is governed by the expectation of a local geometric quantity. We also investigate the approximation of infinite polyhedra by finite ones. We prove that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is necessarily parabolic, revealing a geometric and topological obstruction to the existence of hyperbolic unimodular polyhedral limits. To study stochastic behavior in the hyperbolic regime, we overcome the failure of classical circle packing tools for unbounded degrees by establishing a refined ring lemma for regular circle patterns. This yields effective exponential control of adjacent circle radii via local flower degrees. Combined with boundary methods, we identify the Poisson boundary with the circle at infinity and prove positive hyperbolic speed for the face random walk. These results provide the first quantitative framework connecting local three-dimensional dihedral geometry, global conformal type, and asymptotic stochastic behavior of unimodular random infinite hyperbolic polyhedra.

math.PR↗

arXiv2026

Random infinite ideal angled graphs and ideal hyperbolic polyhedra

This article aims to develop the theory of random infinite ideal hyperbolic polyhedra (abbr. IHP) from multiple perspectives, including combinatorics, geometry, analysis, and random walks. Our starting point is the one-to-one correspondence between IHP and ideal circle packings (ICP), which allows us to translate the theory of IHP into the language of ICP. We then extend the theories of Angel-Hutchcroft-Nachmias-Ray \cite{AHNR16,map} to the ICP setting. This extension is far from straightforward: the presence of dihedral angles introduces substantial new difficulties, requiring new estimates, techniques, and theoretical tools. In particular, we introduce a geometric characteristic number that provides a precise and effective characterization of infinite hyperbolic polyhedra. An IHP $\mathcal P$ corresponds to a weighted planar infinite graph $(G,Θ)$, called an ideal angled graph (abbr. IAG). For unimodular random IAG, we establish an ICP analog of the dichotomy theorem of Angel-Hutchcroft-Nachmias-Ray \cite{AHNR16,map}. Specifically, the geometric characteristic number $T(ρ)=2π-\sum_{e\niρ}Θ_e$ of an IAG determines its ICP type: the graph is a.s. ICP-parabolic iff $\mathbb{E}[T(ρ)]=0$. In the ICP-hyperbolic case, the simple random walk converges a.s. to $\partial\mathbb{D}$ with positive hyperbolic speed. Moreover, the geometric, Poisson, Martin boundaries coincide, extending the boundary theory of Angel-Barlow-Gurevich-Nachmias \cite{ABGN16} and Hutchcroft-Peres \cite{HP17} beyond triangulations to cellular decompositions. As a corollary of the aforementioned IHP/IAG duality, we obtain systematic characterizations of random IHPs.

math.PR↗

arXiv2025

Advanced simulation paradigm of human behaviour unveils complex financial systemic projection

The high-order complexity of human behaviour is likely the root cause of extreme difficulty in financial market projections. We consider that behavioural simulation can unveil systemic dynamics to support analysis. Simulating diverse human groups must account for the behavioural heterogeneity, especially in finance. To address the fidelity of simulated agents, on the basis of agent-based modeling, we propose a new paradigm of behavioural simulation where each agent is supported and driven by a hierarchical knowledge architecture. This architecture, integrating language and professional models, imitates behavioural processes in specific scenarios. Evaluated on futures markets, our simulator achieves a 13.29% deviation in simulating crisis scenarios whose price increase rate reaches 285.34%. Under normal conditions, our simulator also exhibits lower mean square error in predicting futures price of specific commodities. This technique bridges non-quantitative information with diverse market behaviour, offering a promising platform to simulate investor behaviour and its impact on market dynamics.

q-fin.TR↗

arXiv2024

On the Volume Conjecture for hyperbolic Dehn-filled 3-manifolds along the twist knots

For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$ and $|p'|$ for $M$. In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston's triangulation. Our triangulation has led to some new discoveries regarding symmetry, including insights into ``sister manifolds'' as introduced by Hodgson, Meyerhoff, and Weeks.

math.GT↗

arXiv2024

Next-Generation Simulation Illuminates Scientific Problems of Organised Complexity

As artificial intelligence becomes increasingly prevalent in scientific research, data-driven methodologies appear to overshadow traditional approaches in resolving scientific problems. In this Perspective, we revisit a classic classification of scientific problems and acknowledge that a series of unresolved problems remain. Throughout the history of researching scientific problems, scientists have continuously formed new paradigms facilitated by advances in data, algorithms, and computational power. To better tackle unresolved problems, especially those of organised complexity, a novel paradigm is necessitated. While recognising that the strengths of new paradigms have expanded the scope of resolvable scientific problems, we aware that the continued advancement of data, algorithms, and computational power alone is hardly to bring a new paradigm. We posit that the integration of paradigms, which capitalises on the strengths of each, represents a promising approach. Specifically, we focus on next-generation simulation (NGS), which can serve as a platform to integrate methods from different paradigms. We propose a methodology, sophisticated behavioural simulation (SBS), to realise it. SBS represents a higher level of paradigms integration based on foundational models to simulate complex systems, such as social systems involving sophisticated human strategies and behaviours. NGS extends beyond the capabilities of traditional mathematical modelling simulations and agent-based modelling simulations, and therefore, positions itself as a potential solution to problems of organised complexity in complex systems.

cs.AI↗

arXiv2024

Large Language Models Need Consultants for Reasoning: Becoming an Expert in a Complex Human System Through Behavior Simulation

Large language models (LLMs), in conjunction with various reasoning reinforcement methodologies, have demonstrated remarkable capabilities comparable to humans in fields such as mathematics, law, coding, common sense, and world knowledge. In this paper, we delve into the reasoning abilities of LLMs within complex human systems. We propose a novel reasoning framework, termed ``Mosaic Expert Observation Wall'' (MEOW) exploiting generative-agents-based simulation technique. In the MEOW framework, simulated data are utilized to train an expert model concentrating ``experience'' about a specific task in each independent time of simulation. It is the accumulated ``experience'' through the simulation that makes for an expert on a task in a complex human system. We conduct the experiments within a communication game that mirrors real-world security scenarios. The results indicate that our proposed methodology can cooperate with existing methodologies to enhance the reasoning abilities of LLMs in complex human systems.

cs.AI↗

arXiv2023

On uniform estimates for $(n-1)-$form fully nonlinear partial differential equations on compact Hermitian manifolds

We obtain a priori $L^\infty$ estimate for a general class of $(n-1)-$form fully nonlinear partial differential equations on compact Hermitian manifolds. Our method relies on the local version of comparison with auxiliary Monge-Ampère equations, developed earlier by B. Guo and D. H. Phong. The key is to find the appropriate elliptic operator such that the maximum principle applies.

math.AP↗

arXiv2021

On the modulus of continuity of solutions to complex Monge-Ampère equations

In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Ampère equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in Kähler geometry. As an application, we derive a uniform diameter bound for Kähler metrics satisfying certain Monge-Ampère equations.

math.DG↗

arXiv2021

On $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes

The PDE approach developed earlier by the first three authors for $L^\infty$ estimates for fully non-linear equations on Kähler manifolds is shown to apply as well to Monge-Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom-Eyssidieux-Guedj-Zeriahi and Fu-Guo-Song for the Monge-Ampère equation, together with their generalization to Hessian equations.

math.DG↗