Where do hard problems really exist?

Raffaele Marino

Home/Archive/arXiv

arXiv · 2309.16253

Where do hard problems really exist?

Raffaele Marino

View original source ↗Read source document ↗Export metadata ↓

Abstract

This chapter delves into the realm of computational complexity, exploring the world of challenging combinatorial problems and their ties with statistical physics. Our exploration starts by delving deep into the foundations of combinatorial challenges, emphasizing their nature. We will traverse the class P, which comprises problems solvable in polynomial time using deterministic algorithms, contrasting it with the class NP, where finding efficient solutions remains an enigmatic endeavor, understanding the intricacies of algorithmic transitions and thresholds demarcating the boundary between tractable and intractable problems. We will discuss the implications of the P versus NP problem, representing one of the profoundest unsolved enigmas of computer science and mathematics, bearing a tantalizing reward for its resolution. Drawing parallels between combinatorics and statistical physics, we will uncover intriguing interconnections that shed light on the nature of challenging problems. Statistical physics unveils profound analogies with complexities witnessed in combinatorial landscapes. Throughout this chapter, we will discuss the interplay between computational complexity theory and statistical physics. By unveiling the mysteries surrounding challenging problems, we aim to deepen understanding of the very essence of computation and its boundaries. Through this interdisciplinary approach, we aspire to illuminate the intricate tapestry of complexity underpinning the mathematical and physical facets of hard problems.

Explore related subjects

cond-mat.dis-nn cond-mat.stat-mech cs.CC

Keep this discovery

BibTeX RIS

Raffaele Marino. 2023-09-28. Where do hard problems really exist?. https://arxiv.org/abs/2309.16253

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Irrationality Measure Controls Long-Wavelength Charge Fluctuations in Quasiperiodic Systems

Quasiperiodic order is characterized by irrational frequencies whose rational approximability known as irrationality measure defines distinct arithmetic classes. We establish that this arithmetic classification has direct physical consequences for long-wavelength charge fluctuations. In translation-covariant quasiperiodic systems, the infrared scaling of charge fluctuation is governed by the interplay between the irrationality exponent of irrational frequency and the large-harmonic decay of the hull charge profile: the latter determines the available charge weight, while the former controls how efficiently that weight is transferred to the infrared. Consequently, all algebraic irrational frequencies share the same arithmetic scaling, whereas exceptionally well-approximable transcendental frequencies can exhibit strongly enhanced infrared fluctuation scaling. We further prove that occupied states separated from the Fermi level by a gap stable throughout the hull contribute only an analytic infrared background, leaving the nontrivial scaling to near-Fermi states. Our results extend to general translation-covariant multi-frequency quasiperiodic systems.

cond-mat.dis-nn↗

arXiv2026

Curvature-Induced Geometric Universality in Non-Hermitian Anderson Transitions

In Euclidean space, universality classes of Anderson transitions are primarily determined by symmetry and spatial dimensionality. Here, we present evidence for a geometry-controlled universality class of non-Hermitian Anderson transitions on hyperbolic-like lattices. In this setting, critical behavior is influenced by the large-scale hyperbolic geometry, characterized by negative curvature, exponential volume growth, and a non-Euclidean notion of spatial scaling. Finite-size scaling of participation ratios across several distinct $ \{p,q\} $ tilings reveals one-parameter scaling collapses with a common critical exponent $ \nu\simeq1 $ within numerical accuracy. A complementary phenomenological coarse-grained Landau-Ginzburg analysis shows how exponential correlation-volume growth suppresses critical fluctuations, offering a rationale for the observed mean-field-like scaling. Our results suggest that spatial curvature can act as an additional organizing principle for Anderson-transition universality beyond the conventional dimensionality- and symmetry-based classification.

cond-mat.dis-nn↗

arXiv2026

Finite-rank multiplicative perturbations of rotationally invariant non-Hermitian random matrices

We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbation, and $\mathbf{I}$ denotes the identity matrix. We characterize the emergence of outlier eigenvalues, their fluctuations, and the associated eigenvector overlaps. Our results provide a multiplicative non-Hermitian counterpart to the classical Baik--Ben Arous--P\'ech\'e framework.

cond-mat.dis-nn↗