111 research outputs found

    Inhomogeneous quenches as state preparation in two-dimensional conformal field theories

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    The non-equilibrium process where the system does not evolve to the featureless state is one of the new central objects in the non-equilibrium phenomena. In this paper, starting from the short-range entangled state in the two-dimensional conformal field theories (22d CFTs), the boundary state with a regularization, we evolve the system with the inhomogeneous Hamiltonians called M\"obius/SSD ones. Regardless of the details of CFTs considered in this paper, during the M\"obius evolution, the entanglement entropy exhibits the periodic motion called quantum revival. During SSD time evolution, except for some subsystems, in the large time regime, entanglement entropy and mutual information are approximated by those for the vacuum state. We argue the time regime for the subsystem to cool down to vacuum one is t1O(LlA)t_1 \gg \mathcal{O}(L\sqrt{l_A}), where t1t_1, LL, and lAl_A are time, system, and subsystem sizes. This finding suggests the inhomogeneous quench induced by the SSD Hamiltonian may be used as the preparation for the approximately-vacuum state. We propose the gravity dual of the systems considered in this paper, furthermore, and generalize it. In addition to them, we discuss the relation between the inhomogenous quenches and continuous multi-scale entanglement renormalization ansatz (cMERA).Comment: 32+4 pages, 11 figure

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Roth’s Theorem: graph theoretical, analytic and combinatorial proofs

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    In 1936, Erdős–Turán conjectured that any set of integers with positive upper density contains arbitrarily long arithmetic progressions. In 1953, Klaus Roth resolved this conjecture for progressions of length three. This theorem, known as Roth's Theorem, is the main topic of this thesis. In this dissertation we will understand, rewrite and collect some of the proofs of Roth's Theorem that have appeared over the years, while developing some of the problems that arise in each area. This includes the original Fourier analytic proof due to Roth (in a more modern language), the combinatorial proof due to Szemerédi, and finally, the graph theoretical proof based on Szemerédi's Regularity Lemma. We will also explore recent progress around this theorem, as the finite field analogue and the recent breakthrough concerning upper bounds for the cap set problem

    고차 상호작용하는 복잡계의 떠오름 현상

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    학위논문(박사) -- 서울대학교대학원 : 자연과학대학 물리·천문학부(물리학전공), 2022.2. 백용주.구성 요소들의 미시적 동역학만으로는 설명할 수 없는 복잡계의 많은 현상들은 구성 요소들의 복잡한 상호작용에 그 근원을 두고 있다. 총괄적 관점에서 복잡계의 성질을 해석하고 이해하는 방법으로 주목을 받았던 네트워크 과학을 통해 설명해 왔던 물리적 현상들 중에는 짝 상호작용으로 환원되지 않는 성질의 상호작용들이 존재한다. 둘 이상의 많은 요소들의 동시다발적 상호작용을 적절하게 표현하기 위해 사용되는 몇 가지 수학적 도구가 도입되었다. 이 학위논문에서는 고차 상호작용을 포함한 구조로 하이퍼그래프와 단체 복합체를 사용하여 이 위에서 기존 짝 상호작용 구조에서 나타나는 보편성과 동역학들이 어떻게 달라지는지를 해석적, 수치적 방법을 통해 탐구한다. 먼저 고차 상호작용으로의 표현이 필요한 사회 현상인 공저자 네트워크 데이터를 분석한다. 우리는 공저자 네트워크 데이터를 단체 복합체로 표현하여 그 위상적 특성의 시간 진화를 확인하였다. 이것의 성장 과정에서 위상적인 불변량인 베티 수가 차원에 따라 순차적으로 창발이 나타난다는 것을 확인하였다. 이러한 위상적 양상과 구조적 특성을 성장 과정에서 모방할 수 있는 단체 복합체 모형을 구성하여 그 특성들을 통계적으로 확인한 결과, 베티 수의 창발이 무한차수 상전이의 특징을 보인다는 것을 발견하였다. 이를 설명할 수 있는 시간 의존 비율방정식의 정상상태를 생성함수 방법으로 접근하여 무한차수 상전이를 이론적으로 규명하였다. 하이퍼그래프는 사회 연결망에서 여러 사람이 함께 하는 팀 사이의 상호작용을 표현하는데에도 사용할 수 있다. 매개 중심성을 하이퍼그래프에서 측정할 전산 알고리즘을 제안하고, 사회 연결망을 묘사할 척도없는 하이퍼그래프 모형을 구성하였다. 이로부터 팀의 매개중심성 분포는 개개인의 매개중심성 분포와는 상이하게 거듭제곱 분포가 왜곡되어 지수함수적 감소를 보임을 확인하고, 이 감소의 정도가 팀의 크기와 관련됨을 보았다. 하지만 실제 데이터에 반영되는 추가 정보를 팀의 가중치로써 도입하면 매개중심성 분포의 거듭제곱 법칙이 재구성됨을 확인하였다. 하지만 팀이 가지는 가중치가 아니라 하이퍼그래프 구조에서 존재하는 위치가 팀의 성과에 더욱 중요한 요소라는 반직관적인 결과를 얻었다. 마지막으로 복잡계에서 보이는 동역학 과정에 미치는 고차 상호작용의 영향을 동기화 현상의 예를 통해 살펴보았다. 동기화 과정은 자연 및 인공 시스템의 광범위한 기능에 중요한 역할을 하기 때문에 그 집단 행동의 형성에 미치는 미시적 상호작용 구조에 대한 이해는 필수불가결하다고 할 수 있다. 우리는 척도없는 하이퍼그래프 위에서의 구라모토 모형을 연속 방정식의 오트-안톤센 가설 풀이법과 평균장 이론을 이용하여 조사하였다. 그 결과로 연결 구조의 불균일함에 따라 연속 상전이에서 불연속 상전이로의 변화가 나타남을 확인하였다. 특히 불연속 상전이는 그 전이점에서 임계 현상을 보이는 하이브리드 상전이임을 확인하였고, 이 임계 지수를 해석적, 수치적으로 결정하였다.Rooted in the complex interactions of components, diverse aspects in complex systems cannot be explained by the microscopic dynamics of components. Among the physical phenomena that have been described through network science, which have received attention as a way of interpreting and understanding the properties of complex systems from a holistic point of view, are intrinsic interactions that cannot be reduced to pairwise interactions. In this dissertation, we analytically and numerically explore the emergent phenomena that appear in these higher-order interactions. As structures include these simultaneous interactions of two or more elements, we use hypergraphs and simplicial complexes. First, the coauthorship data, which is a social relationship that requires the expression of higher-order interactions, is analyzed. We confirmed the time evolution of the topological features by expressing the coauthorship data as a simplicial complex. In the process of its growth, we find that the Betti numbers, topological invariants, sequentially appeared according to the dimension. As a result of statistical confirming the properties by constructing a random simplicial complex model that can imitate these topological aspects and structural characteristics in the growing process, we reveal that the development of the Betti number showed the infinite-order phase transition. By generating function, the steady-state analysis of the time-dependent rate equation that mimics the model algorithms suggests the theoretical explanations of the infinite-order transitions. Hypergraphs also are widely used to express interactions between teams with multiple people in a social network. Here we propose a computational algorithm to measure betweenness centralities in a hypergraph. Furthermore, a scale-free hypergraph model is constructed to describe the features of social networks. From this, we confirm that the distribution of the team's betweenness centrality, showing exponential decaying, is different from the power-law distribution of individual betweenness centrality. Interestingly, the decaying rate is related to the size of the team. However, if additional information reflected in the actual data is introduced as the team's weight, the power law of the betweenness centrality distribution is reconstructed. Counterintuitively, we observe that the location of a team in the hypergraph structure, such as whether the team is near the hub, not the weight of the team, is a more crucial factor in the team's performance. Finally, the influence of higher-order interactions on the dynamics detected in complex systems is examined through examples of synchronization phenomena. Synchronization appears in a wide range of natural and artificial systems. Thus, an understanding of the microscopic group interaction structure is indispensable. We investigated the Kuramoto model on a scaleless hypergraph using the Ott-Antonsen ansatz of continuity equations and the mean-field theory. As a result, we find that the non-uniformity of the connection structure resulted in a change from continuous phase transition to discontinuous phase transition. In particular, we observe that discontinuous phase transition is indeed a hybrid phase transition, showing a critical phenomenon at its transition point. The critical exponents are determined both analytically and numerically.Abstract i Contents iii List of Figures vii List of Tables ix 1 Introduction 1 1.1 Complex systems in nature 1 1.2 Representations of complex systems 3 1.3 Structure and goal of the dissertation 6 2 Representations of interactions 10 2.1 Interaction 10 2.2 Pairwise represenation: graph theory 11 2.2.1 Definitions and concepts 11 2.2.2 Random graph models 17 2.3 Higher-order representation: concepts of hypergraphs 19 2.3.1 Mathematical definitions 19 2.4 Algebraic topology: special case of higher-order interactions 21 2.4.1 Simplicial complexes 21 2.4.2 Homology groups 22 3 Basics of percolation transitions 25 3.1 What is percolation 25 3.2 Percolation on complex networks 27 4 Homological percolation transitions: numerical approach 29 4.1 Introduction 29 4.2 Results 33 4.2.1 Homological percolation transitions 33 4.2.2 Facet degree distribution 37 4.2.3 Minimal model 39 4.2.4 Kahle localization 42 4.3 Conclusion 44 4.4 Discussion 45 5 Homological percolation transitions: analytical approach 46 5.1 Introduction 46 5.2 Model 49 5.3 Percolation transition 49 5.3.1 Cluster-size distribution, giant cluster, and mean cluster size 49 5.3.2 Graph and facet degree distributions 55 5.4 Homological percolation transition 58 5.4.1 Model generalization and Betti number 58 5.4.2 Rigorous description of the first Betti number 60 5.5 Discussion 63 5.6 Conclusion 64 6 Criticality from shortest path dynamics on hypergraphs: Betweenness centraility distribution 65 6.1 Betweenness centrality in hypergraphs 65 6.2 Random hypergraph model with preferential attachment 68 6.3 Real data analysis 72 6.4 Summary and discussion 76 7 Synchronization of coupled oscillators 81 7.1 Synchronization 81 7.2 Kuramoto model 82 7.2.1 Order parameter 83 7.2.2 Self-consistency equation approaches 84 8 Synchronization transitions in hypergraphs 86 8.1 Introduction 86 8.2 Model 87 8.3 Self-consistency equations 89 8.4 Results 92 8.5 Summary and discussion 92 9 Conclusion 95 Appendices 97 Appendix A Topological data analysis 98 A.1 Persistent homology 98 Appendix B Appendices for chapter 4 100 B.1 Derivation of master equation of joint generating function 100 Appendix C Appendices for chapter 5 103 C.1 BC distributions for the BA-II hypergraphs 103 C.2 Features of our coauthorship data 104 C.3 Parameter analysis 105 C.3.1 Relations among the BCes and parameters 105 C.3.2 Influential parameters for top BCes 107 C.4 Analysis of the small size dataset 108 Appendix D Appendices for chapter 6 118 D.1 Heterogeneous mean-field theory 118 D.2 Critical behavior 120 D.3 Correlation size 121 Bibliography 123 Abstract in Korean 135박

    Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics

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    Synchronization in Complex Networks Under Uncertainty

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    La sincronització en xarxes és la música dels sistemes complexes. Els ritmes col·lectius que emergeixen de molts oscil·ladors acoblats expliquen el batec constant del cor, els patrons recurrents d'activitat neuronal i la sincronia descentralitzada a les xarxes elèctriques. Els models matemàtics són sòlids i han avançat significativament, especialment en el problema del camp mitjà, on tots els oscil·ladors estan connectats mútuament. Tanmateix, les xarxes reals tenen interaccions complexes que dificulten el tractament analític. Falta un marc general i les soluciones existents en caixes negres numèriques i espectrals dificulten la interpretació. A més, la informació obtinguda en mesures empíriques sol ser incompleta. Motivats per aquestes limitacions, en aquesta tesi proposem un estudi teòric dels oscil·ladors acoblats en xarxes sota incertesa. Apliquem propagació d'errors per predir com una estructura complexa amplifica el soroll des dels pesos microscòpics fins al punt crític de sincronització, estudiem l'efecte d'equilibrar les interaccions de parelles i d'ordre superior en l'optimització de la sincronia i derivem esquemes d'ajust de pesos per mapejar el comportament de sincronització en xarxes diferents. A més, un desplegament geomètric rigorós de l'estat sincronitzat ens permet abordar escenaris descentralitzats i descobrir regles locals òptimes que indueixen transicions globals abruptes. Finalment, suggerim dreceres espectrals per predir punts crítics amb àlgebra lineal i representacions aproximades de xarxa. En general, proporcionem eines analítiques per tractar les xarxes d'oscil·ladors en condicions sorolloses i demostrem que darrere els supòsits predominants d'informació completa s'amaguen explicacions mecanicistes clares. Troballes rellevants inclouen xarxes particulars que maximitzen el ventall de comportaments i el desplegament exitós del binomi estructura-dinàmica des d'una perspectiva local. Aquesta tesi avança la recerca d'una teoria general de la sincronització en xarxes a partir de principis mecanicistes i geomètrics, una peça clau que manca en l'anàlisi, disseny i control de xarxes neuronals biològiques i artificials i sistemes d'enginyeria complexos.La sincronización en redes es la música de los sistemas complejos. Los ritmos colectivos que emergen de muchos osciladores acoplados explican el latido constante del corazón, los patrones recurrentes de actividad neuronal y la sincronía descentralizada de las redes eléctricas. Los modelos matemáticos son sólidos y han avanzado significativamente, especialmente en el problema del campo medio, donde todos los osciladores están conectados entre sí. Sin embargo, las redes reales tienen interacciones complejas que dificultan el tratamiento analítico. Falta un marco general y las soluciones en cajas negras numéricas y espectrales dificultan la interpretación. Además, las mediciones empíricas suelen ser incompletas. Motivados por estas limitaciones, en esta tesis proponemos un estudio teórico de osciladores acoplados en redes bajo incertidumbre. Aplicamos propagación de errores para predecir cómo una estructura compleja amplifica el ruido desde las conexiones microscópicas hasta puntos críticos macroscópicos, estudiamos el efecto de equilibrar interacciones por pares y de orden superior en la optimización de la sincronía y derivamos esquemas de ajuste de pesos para mapear el comportamiento en estructuras distintas. Una expansión geométrica del estado sincronizado nos permite abordar escenarios descentralizados y descubrir reglas locales que inducen transiciones abruptas globales. Por último, sugerimos atajos espectrales para predecir puntos críticos usando álgebra lineal y representaciones aproximadas de red. En general, proporcionamos herramientas analíticas para manejar redes de osciladores en condiciones ruidosas y demostramos que detrás de las suposiciones predominantes de información completa se ocultaban claras explicaciones mecanicistas. Hallazgos relevantes incluyen redes particulares que maximizan el rango de comportamientos y la explicación del binomio estructura-dinámica desde una perspectiva local. Esta tesis avanza en la búsqueda de una teoría general de sincronización en redes desde principios mecánicos y geométricos, una pieza clave que falta en el análisis, diseño y control de redes neuronales biológicas y artificiales y sistemas de ingeniería complejos.Synchronization in networks is the music of complex systems. Collective rhythms emerging from many interacting oscillators appear across all scales of nature, from the steady heartbeat and the recurrent patterns in neuronal activity to the decentralized synchrony in power-grids. The mathematics behind these processes are solid and have significantly advanced lately, especially in the mean-field problem, where oscillators are all mutually connected. However, real networks have complex interactions that difficult the analytical treatment. A general framework is missing and most existing results rely on numerical and spectral black-boxes that hinder interpretation. Also, the information obtained from measurements is usually incomplete. Motivated by these limitations, in this thesis we propose a theoretical study of network-coupled oscillators under uncertainty. We apply error propagation to predict how a complex structure amplifies noise from the link weights to the synchronization onset, study the effect of balancing pair-wise and higher-order interactions in synchrony optimization, and derive weight-tuning schemes to map the synchronization behavior of different structures. Also, we develop a rigorous geometric unfolding of the synchronized state to tackle decentralized scenarios and to discover optimal local rules that induce global abrupt transitions. Last, we suggest spectral shortcuts to predict critical points using linear algebra and network representations with limited information. Overall, we provide analytical tools to deal with oscillator networks under noisy conditions and prove that mechanistic explanations were hidden behind the prevalent assumptions of complete information. Relevant finding include particular networks that maximize the range of behaviors and the successful unfolding of the structure-dynamics interplay from a local perspective. This thesis advances the quest of a general theory of network synchronization built from mechanistic and geometric principles, a key missing piece in the analysis, design and control of biological and artificial neural networks and complex engineering systems

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum
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