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Haoran Zhu

Publications and source records attributed to Haoran Zhu.

At least 19 recordsLinked to original sources
arXiv2026

The maximum relaxation time of a random walk on regular graphs

We determine the asymptotic maximum relaxation time of simple random walk on connected simple regular graphs of a given order. The leading constant depends on the parity of the order: cubic graphs are asymptotically extremal at even orders, whereas quartic graphs are asymptotically extremal at odd orders. We obtain an asymptotically sharp upper bound uniform in the degree; this proves the longstanding Aldous--Fill spectral gap conjecture. A uniform strict improvement for noncubic graphs implies that, for every sufficiently large even order, the maximum is attained uniquely by the cubic graph of least algebraic connectivity. We also prove quantitative stability for cubic near-extremisers. For prescribed edge-connectivity, we determine the sharp bound when the degree tends to infinity and characterise asymptotic equality.

math.PR↗
arXiv2026

Universal Pl\"ucker positivity and the Octopus inequality

We prove a positivity theorem for the universal Pl\"ucker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised $(2,2)$-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Pl\"ucker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.

math.RT↗
arXiv2026

Random walks on wreath products and spectral gaps for coloured interchange processes

We introduce group-valued coloured interchange processes, a class of continuous-time random walks on wreath products generated by transpositions and arbitrary symmetric base-group transitions. We give a complete representation-theoretic characterisation of their spectral gaps by decomposing the associated quasi-regular representation and determining precisely which irreducible representations are required. For abelian base groups, we realise these representations on multislices and identify their Laplacians with discrete Schr\"odinger operators. We then classify the minimal families of irreducible representations that determine the spectral gap for every choice of transition rates.

math.PR↗
arXiv2026

iSWAP maximises the second-moment spectral gap in random quantum circuits

We prove that the $\mathrm{iSWAP}$ gate maximises the spectral gap of the Hermitian second-moment operator on every connected graph with at least three vertices, among all two-local unitary circuit ensembles. We further prove that the polyhedral cone defined by asymmetric four-point inequalities is invariant under the transpose of the $\mathrm{iSWAP}$ semigroup, yielding a componentwise comparison certificate for its positive Perron--Frobenius eigenvector. These results resolve a conjecture of Kong, Li, and Liu.

quant-ph↗
arXiv2026

Spectral gaps of ironed two-qubit gadgets matching the iSWAP gap

We prove that every ironed two-qubit gadget whose KAK-derived parameter satisfies $a=5/9$ has, on the complete graph $K_n$ with $n\geqslant 5$, the same second-moment spectral gap as the iSWAP gadget. The central step is a representation-theoretic localisation theorem: the largest strictly negative eigenvalue of the associated $\mathfrak S_n$-invariant operator always occurs in the highest-spin $\mathrm{SU}(2)$ summand. A local positive-semidefinite decomposition separates every spin sector except the two highest. This settles a conjecture of Kong, Li, and Liu.

quant-ph↗
arXiv2026

PBW bases and centralisers for the $q$-Onsager algebra

We prove that, over an arbitrary field and whenever $q$ is not a root of unity, the Baseilhac--Kolb root vectors form a PBW basis of the $q$-Onsager algebra for every total order on the positive roots of $\widehat{\mathfrak{sl}}_2$. This removes the previous transcendence hypothesis. We establish twelve PBW bases in the alternating generators and show that they persist under arbitrary scalar central specialisation of the alternating central extension. We determine the centraliser of the negative alternating subalgebra and that of the first imaginary alternating generator, and deduce that the four single-family alternating polynomial subalgebras are maximal commutative. Together, these results settle four conjectures of Terwilliger and, in characteristic different from $2$, a conjecture of Baseilhac and Belliard.

math-ph↗
arXiv2026

A new PBW basis for the alternating central extension of the $q$-Onsager algebra

We establish a new PBW basis for $\mathcal A_q$, the alternating central extension of the $q$-Onsager algebra. Terwilliger showed that the alternating generators form a PBW basis in the block order $\mathcal G<\mathcal W^-<\mathcal W^+<\widetilde{\mathcal G}$. We prove that they also form a PBW basis in the different block order $\mathcal W^-<\mathcal G<\widetilde{\mathcal G}<\mathcal W^+$. Consequently, multiplication induces a vector-space isomorphism \[ \mathcal W^-\otimes\mathcal G\otimes\widetilde{\mathcal G}\otimes\mathcal W^+ \longrightarrow \mathcal A_q, \] thereby confirming a conjecture of Terwilliger.

math-ph↗
arXiv2026

RenderFormer++: Scalable and Physics-Informed Feed-Forward Neural Rendering

We present RenderFormer++, a scalable and physics-informed feed-forward neural rendering framework for global illumination in mesh scenes. Existing Transformer-based neural rendering methods such as RenderFormer achieve promising cross-scene generalization, but lack explicit transport priors and scale poorly due to quadratic triangle-level attention. To address these issues, we introduce Physics-Informed Transport Guidance (PITG), which embeds rendering-equation-inspired inductive biases into the attention mechanism and introduces a transport consistency loss, encouraging physics-informed light transport modeling. We further propose Hierarchical Object-Centric Tokenization (HOCT), which aggregates triangle-level features into compact object-level tokens via cross-attention with learnable queries, substantially reducing computational and memory costs. Extensive experiments demonstrate that RenderFormer++ achieves scalable and generalizable feed-forward global illumination rendering across complex large-scale scenes with competitive rendering quality and substantially improved efficiency over RenderFormer. The code will be made publicly available upon acceptance.

cs.GR↗
arXiv2026

Mesh2GS: White-Box 3DGS Construction via Plenoptic Sampling

3D Gaussian Splatting (3DGS) has emerged as a promising method for high-quality, real-time 3D reconstruction. To associate 3DGS with mesh representations, existing methods primarily focus on 3DGS-to-mesh reconstruction from multi-view images. In contrast, the problem of converting a mesh into 3DGS has received comparatively less attention. Instead of relying on heuristic strategies that bind 3D Gaussians to the mesh, we propose a novel white-box 3DGS construction framework, termed Mesh2GS, which generates 3DGS directly from mesh geometry based on plenoptic sampling theory, achieving Nyquist-level performance for high-quality global illumination rendering. Firstly, we propose a plenoptic sampling guided 3DGS construction strategy that theoretically derives the minimum sampling rate of the sampled views and the distribution of 3D Gaussians. Second, we propose a novel 3DGS update procedure with albedo--shading decomposition for efficient global-illumination capture. Finally, we introduce a neural illumination enhancement module to handle non-Lambertian effects. Experimental results demonstrate that our method surpasses state-of-the-art baselines and is practically effective for both real-time shared rendering and non-Lambertian effects capturing specular highlights. The project code will be released upon acceptance.

cs.GR↗
arXiv2026

Zero-Shot Cross-City Generalization in End-to-End Autonomous Driving: Self-Supervised versus Supervised Representations

End-to-end autonomous driving models are typically trained on multi-city datasets using supervised ImageNet-pretrained backbones, yet their ability to generalize to unseen cities remains largely unexamined. When training and evaluation data are geographically mixed, models may implicitly rely on city-specific cues, masking failure modes that would occur under real-world domain shifts when generalizing to new locations. In this work, we formulate zero-shot cross-city transfer as a controlled representation-level stress test for end-to-end autonomous driving and ask how visual pretraining affects transfer behavior under geographic domain shift. We conduct a comprehensive study by integrating self-supervised backbones I-JEPA, DINOv2, and MAE into planning frameworks. We evaluate performance under strict geographic splits on nuScenes in the open-loop setting and on NAVSIM in the closed-loop evaluation protocol. Our experiments reveal a substantial generalization gap when transferring models across cities with different road topologies, traffic conventions, and visual environments. In open-loop evaluation, a supervised backbone exhibits severe degradation when transferring between cities, yet some domain-specific self-supervised methods can substantially reduce both displacement and collision degradation. In closed-loop evaluation, self-supervised pretraining improves average out-of-distribution PDMS in several single-city training settings. Our results provide empirical evidence that representation learning influences the robustness of cross-city planning and motivate zero-shot geographic transfer as an important stress test for evaluating end-to-end autonomous driving systems.

cs.CV↗
arXiv2026

The Hurwitz sum-of-squares problem depends on the base field

We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type $[12,12,18]$ over every field of characteristic different from $2$ in which $-1$ is a square, whereas no such formula exists over any formally real field. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$. This settles, in the negative, a longstanding conjecture of Shapiro from 1984, a conjecture of Adem from 1975, and answers a signed-formula problem raised by Shapiro in 2000.

math.NT↗
arXiv2026

A proof of Haemers' toughness conjecture

We prove that if $Γ$ is a connected graph with minimum degree $δ$ and Laplacian eigenvalues $0=μ_1<μ_2\leqslant \cdots \leqslant μ_n$, then the toughness of $Γ$ is bounded below by $μ_2/(μ_n-δ)$.

math.CO↗
arXiv2026

AcademiClaw: When Students Set Challenges for AI Agents

Benchmarks within the OpenClaw ecosystem have thus far evaluated exclusively assistant-level tasks, leaving the academic-level capabilities of OpenClaw largely unexamined. We introduce AcademiClaw, a bilingual benchmark of 80 complex, long-horizon tasks sourced directly from university students' real academic workflows -- homework, research projects, competitions, and personal projects -- that they found current AI agents unable to solve effectively. Curated from 230 student-submitted candidates through rigorous expert review, the final task set spans 25+ professional domains, ranging from olympiad-level mathematics and linguistics problems to GPU-intensive reinforcement learning and full-stack system debugging, with 16 tasks requiring CUDA GPU execution. Each task executes in an isolated Docker sandbox and is scored on task completion by multi-dimensional rubrics combining six complementary techniques, with an independent five-category safety audit providing additional behavioral analysis. Experiments on six frontier models show that even the best achieves only a 55\% pass rate. Further analysis uncovers sharp capability boundaries across task domains, divergent behavioral strategies among models, and a disconnect between token consumption and output quality, providing fine-grained diagnostic signals beyond what aggregate metrics reveal. We hope that AcademiClaw and its open-sourced data and code can serve as a useful resource for the OpenClaw community, driving progress toward agents that are more capable and versatile across the full breadth of real-world academic demands. All data and code are available at https://github.com/GAIR-NLP/AcademiClaw.

cs.AI↗
arXiv2026

UHR-DETR: Efficient End-to-End Small Object Detection for Ultra-High-Resolution Remote Sensing Imagery

Ultra-High-Resolution (UHR) imagery has become essential for modern remote sensing, offering unprecedented spatial coverage. However, detecting small objects in such vast scenes presents a critical dilemma: retaining the original resolution for small objects causes prohibitive memory bottlenecks. Conversely, conventional compromises like image downsampling or patch cropping either erase small objects or destroy context. To break this dilemma, we propose UHR-DETR, an efficient end-to-end transformer-based detector designed for UHR imagery. First, we introduce a Coverage-Maximizing Sparse Encoder that dynamically allocates finite computational resources to informative high-resolution regions, ensuring maximum object coverage with minimal spatial redundancy. Second, we design a Global-Local Decoupled Decoder. By integrating macroscopic scene awareness with microscopic object details, this module resolves semantic ambiguities and prevents scene fragmentation. Extensive experiments on the UHR imagery datasets (e.g., STAR and SODA-A) demonstrate the superiority of UHR-DETR under strict hardware constraints (e.g., a single 24GB RTX 3090). It achieves a 2.8\% mAP improvement while delivering a 10$\times$ inference speedup compared to standard sliding-window baselines on the STAR dataset. Our codes and models will be available at https://github.com/Li-JingFang/UHR-DETR.

cs.CV↗
arXiv2026

Spectral gap of biased adjacent-transposition chains

We establish a sharp lower bound on the spectral gap of the biased adjacent-transposition Markov chain on the symmetric group. As a consequence, we resolve a longstanding conjecture of Fill, proving that among all regular probability vectors, the minimum spectral gap of the transition matrix is attained by the uniform probability vector. We also characterise the regular probability vectors attaining the minimum spectral gap and determine the exact multiplicity of the corresponding second-largest eigenvalue. Our proof relies on a novel algebraic decomposition of the transition matrix into elementary orthogonal projections.

math.PR↗
arXiv2026

Generalized Small Object Detection:A Point-Prompted Paradigm and Benchmark

Small object detection (SOD) remains challenging due to extremely limited pixels and ambiguous object boundaries. These characteristics lead to challenging annotation, limited availability of large-scale high-quality datasets, and inherently weak semantic representations for small objects. In this work, we first address the data limitation by introducing TinySet-9M, the first large-scale, multi-domain dataset for small object detection. Beyond filling the gap in large-scale datasets, we establish a benchmark to evaluate the effectiveness of existing label-efficient detection methods for small objects. Our evaluation reveals that weak visual cues further exacerbate the performance degradation of label-efficient methods in small object detection, highlighting a critical challenge in label-efficient SOD. Secondly, to tackle the limitation of insufficient semantic representation, we move beyond training-time feature enhancement and propose a new paradigm termed Point-Prompt Small Object Detection (P2SOD). This paradigm introduces sparse point prompts at inference time as an efficient information bridge for category-level localization, enabling semantic augmentation. Building upon the P2SOD paradigm and the large-scale TinySet-9M dataset, we further develop DEAL (DEtect Any smalL object), a scalable and transferable point-prompted detection framework that learns robust, prompt-conditioned representations from large-scale data. With only a single click at inference time, DEAL achieves a 31.4% relative improvement over fully supervised baselines under strict localization metrics (e.g., AP75) on TinySet-9M, while generalizing effectively to unseen categories and unseen datasets. Our project is available at https://zhuhaoraneis.github.io/TinySet-9M/.

cs.CV↗
arXiv2026

A symmetry formula for correlation functions in the superintegrable chiral Potts spin chain

We prove an exact finite-volume symmetry formula for two-point functions in the periodic $N$-state superintegrable chiral Potts spin chain. We show that, for every chain length $L$ and every simultaneous eigenvector of the Hamiltonian and the one-site translation operator, the correlations satisfy $\langle Z_0^r Z_R^{\dagger r}\rangle^*=\langle Z_0^r Z_{L-R}^{\dagger r}\rangle$ for $1\leqslant r\leqslant N-1$. Hence, whenever $L$ is even, the midpoint correlation $\langle Z_0^r Z_{L/2}^{\dagger r}\rangle$ is real. Then we generalise the three-state chain case to arbitrary $N$ and to every translation eigensector. This resolves a conjecture of Fabricius and McCoy.

math-ph↗
arXiv2026

Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity

We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad α_t = \mathrm{id} + tα_1 \] define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,α_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010.

math.RA↗