The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulas relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relationship between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.

The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian related to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formules relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. En particulier, nous dérive une relation intégrale between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we calcul the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.

The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinguished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulas relating the potential defining a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. In particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compute the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.

The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. As such, the main goal of random matrix theory (RMT) has been to derive the eigenvalue statistics of matrices drawn from a given distribution. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. In the resulting matrix, the diagonal and off-diagonal Lanczos coefficients control transition amplitudes between elements of a distinished basis of states. We connect these two approaches to the quantum mechanics of complex systems by deriving analytical formulas relating the potential definiing a general RMT, or, equivalently, its density of states, to the Lanczos coefficients and their correlations. En particular, we derive an integral relation between the average Lanczos coefficients and the density of states, and, for polynomial potentials, algebraic equations that determine the Lanczos coefficients from the potential. We obtain these results for generic initial states in the thermodynamic limit. As an application, we compuute the time-dependent ``spread complexity'' in thermofield double states and the spectral form factor for Gaussian and non-Gaussian RMTs.

قدم الفيزيائي المجري يوجين ويغنر نماذج مصفوفة عشوائية في الفيزياء لوصف أطياف الطاقة للنوى الذرية. على هذا النحو، كان الهدف الرئيسي لنظرية المصفوفة العشوائية (rmt) هو اشتقاق إحصائيات القيمة الذاتية للمصفوفات المستمدة من توزيع معين. يعطي نهج Wigner رؤى قوية حول خصائص الأنظمة المعقدة والفوضوية في التوازن الحراري. اقترح مجري آخر، كورنيليوس لانزوس، طريقة لتقليل ديناميكيات أي نظام كمي إلى سلسلة أحادية البعد عن طريق تحويل هاميلتون إلى حالة أولية معينة. في المصفوفة الناتجة، تتحكم معاملات لانزوس القطرية وغير القطرية في سعة الانتقال بين عناصر أساس مميز من الحالات. نحن نربط هذين النهجين بميكانيكا الكم للأنظمة المعقدة من خلال اشتقاق الصيغ التحليلية التي تربط الإمكانات التي تحدد rmt العام، أو، على نحو مكافئ، كثافته من الحالات، بمعاملات لانزوس وارتباطاتها. على وجه الخصوص، نستمد علاقة متكاملة بين متوسط معاملات Lanczos وكثافة الحالات، وبالنسبة لجهد متعدد الحدود، المعادلات الجبرية التي تحدد معاملات Lanczos من الجهد. نحصل على هذه النتائج للحالات الأولية العامة في الحد الديناميكي الحراري. كتطبيق، نحسب "تعقيد الانتشار" المعتمد على الوقت في الحالات المزدوجة للحقل الحراري وعامل الشكل الطيفي لـ RMTs الغاوسية وغير الغاوسية.