TITLES & ABSTRACTS
Alessia Annibale: Non-equilibrium dynamics of Sparse Boolean Networks
The dynamic cavity method provides an efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. However, the complexity of the cavity approach grows exponentially with the in-degrees of the interacting units, which creates a de-facto barrier in systems with fat-tailed in-degree distributions.
We present a dynamic programming algorithm that reduces the computational complexity from exponential to quadratic, whenever couplings are chosen randomly from a discrete set of equidistant values. As a case study, we analyse the heterogeneous statistics of single node activation in random Boolean networks with fat-tailed degree distribution and fully asymmetric binary couplings. In addition, we extend the dynamical cavity approach to calculate the pairwise correlations induced by different motifs in the network. Our results suggest that the statistics of observed correlations can be accurately described in terms of two basic motifs.
We then investigate models with sparse, bi-directional interactions. We observe that the stationary state associated with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.
Our study shows that a degree of bi-directionality in the interactions is conducive to having multiple attractors, when noise is sufficiently low, and the presence of multi-node interactions increases the diversity of attractors. These facts may hint at the mechanism with which gene regulatory networks sustain multi-cellular life.
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Thibault Bonnemain: Quadratic mean field games: Schrödinger and electrostatic representations
Mean Field Games provide a powerful theoretical framework to deal with stochastic optimization problems involving a large number of coupled subsystems. They can find application in several fields, be it finance, economics, sociology, engineering ... Though these models are much simpler than the underlying differential games they describe in some limit, their behaviour is not yet fully understood. I will focus on a toy models from a particular class of games, for which there is a deep connection between the associated system of PDEs and the nonlinear Schrödinger equation. This allows me to identify limiting regimes that can be dealt with, in particular one such regime yields insight on the intrinsic forward-backward structure of mean field games through a mapping to an electrostatic problem.
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Luca Capizzi: Entanglement evolution after a global quench across a conformal defect
We consider the dynamics of a one-dimensional quantum system in the presence of a localized defect. We prepare the system in a short-range entangled state, we let it evolve ballistically, and we study the entanglement across the defect. Linear growth of the entanglement entropy is observed, whose slope depends both on the scattering properties of the defect and the initial state. The protocol above is characterized in Conformal Field Theory, and the Rényi entropies are related to the correlation functions of twist fields in a bounded two-dimensional geometry. Moreover, we investigate a particular lattice realization in a free-fermion chain, giving a prediction for the linear slope via a quasi-particle picture. This talk is based on joint work with Viktor Eisler.
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Giuseppe Del Vecchio Del Vecchio: Generalised hydrodynamics and correlation functions
While statistical mechanics at equilibrium is relatively well settled, out-of-equilibrium phenomena escape universal description and represent one of the main challenges of theoretical physics nowadays. In this directions, in the past couple of decades, integrable models have represented a major playground territory not only because integrability gives access to analytic predictions but mainly because standard laboratory practices can now put them under direct experimental scrutiny. The main consequence of integrability is the presence of infinitely many conservation laws which strongly constrain the dynamics making the equilibrium ensembles rather different than the ordinary Gibbs state observed in presence of generic interactions. Generalised hydrodynamics (GHD) is the theory capturing long wavelength and large spacetime universal behaviour of integrable models and in the past five years it has produced an incredible amount of results in the field of our-of-equilibrium physics. In this contribution I will introduce GHD and discuss how, in combination with large deviation theory, it gives access to exact Euler scale (and beyond) correlation functions. Applications to XX and XY spin chain spin-spin correlation function and calculation of Renyi’s and entanglement entropy are presented.
_______________________________________________________________________________________________
Katja Koblas: Growth of Rényi entropies in interacting integrable models
After a quantum quench in a clean quantum many-body system Rényi entropies generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out-of-equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system's dynamics, and its characterisation is a key objective of current research. In the talk I will show that the slope of Rényi entropies can be determined by means of a spacetime duality transformation. This allows for an explicit exact formula for the slope of Rényi entropies in all integrable models treatable by thermodynamic Bethe ansatz. Interestingly, the formula can be understood in terms of a quasiparticle picture only in the von Neumann limit. The talk is based on Phys. Rev. X 12 031016 (2022).
Alessia Annibale: Non-equilibrium dynamics of Sparse Boolean Networks
The dynamic cavity method provides an efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. However, the complexity of the cavity approach grows exponentially with the in-degrees of the interacting units, which creates a de-facto barrier in systems with fat-tailed in-degree distributions.
We present a dynamic programming algorithm that reduces the computational complexity from exponential to quadratic, whenever couplings are chosen randomly from a discrete set of equidistant values. As a case study, we analyse the heterogeneous statistics of single node activation in random Boolean networks with fat-tailed degree distribution and fully asymmetric binary couplings. In addition, we extend the dynamical cavity approach to calculate the pairwise correlations induced by different motifs in the network. Our results suggest that the statistics of observed correlations can be accurately described in terms of two basic motifs.
We then investigate models with sparse, bi-directional interactions. We observe that the stationary state associated with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.
Our study shows that a degree of bi-directionality in the interactions is conducive to having multiple attractors, when noise is sufficiently low, and the presence of multi-node interactions increases the diversity of attractors. These facts may hint at the mechanism with which gene regulatory networks sustain multi-cellular life.
_______________________________________________________________________________________________
Thibault Bonnemain: Quadratic mean field games: Schrödinger and electrostatic representations
Mean Field Games provide a powerful theoretical framework to deal with stochastic optimization problems involving a large number of coupled subsystems. They can find application in several fields, be it finance, economics, sociology, engineering ... Though these models are much simpler than the underlying differential games they describe in some limit, their behaviour is not yet fully understood. I will focus on a toy models from a particular class of games, for which there is a deep connection between the associated system of PDEs and the nonlinear Schrödinger equation. This allows me to identify limiting regimes that can be dealt with, in particular one such regime yields insight on the intrinsic forward-backward structure of mean field games through a mapping to an electrostatic problem.
_______________________________________________________________________________________________
Luca Capizzi: Entanglement evolution after a global quench across a conformal defect
We consider the dynamics of a one-dimensional quantum system in the presence of a localized defect. We prepare the system in a short-range entangled state, we let it evolve ballistically, and we study the entanglement across the defect. Linear growth of the entanglement entropy is observed, whose slope depends both on the scattering properties of the defect and the initial state. The protocol above is characterized in Conformal Field Theory, and the Rényi entropies are related to the correlation functions of twist fields in a bounded two-dimensional geometry. Moreover, we investigate a particular lattice realization in a free-fermion chain, giving a prediction for the linear slope via a quasi-particle picture. This talk is based on joint work with Viktor Eisler.
_______________________________________________________________________________________________
Giuseppe Del Vecchio Del Vecchio: Generalised hydrodynamics and correlation functions
While statistical mechanics at equilibrium is relatively well settled, out-of-equilibrium phenomena escape universal description and represent one of the main challenges of theoretical physics nowadays. In this directions, in the past couple of decades, integrable models have represented a major playground territory not only because integrability gives access to analytic predictions but mainly because standard laboratory practices can now put them under direct experimental scrutiny. The main consequence of integrability is the presence of infinitely many conservation laws which strongly constrain the dynamics making the equilibrium ensembles rather different than the ordinary Gibbs state observed in presence of generic interactions. Generalised hydrodynamics (GHD) is the theory capturing long wavelength and large spacetime universal behaviour of integrable models and in the past five years it has produced an incredible amount of results in the field of our-of-equilibrium physics. In this contribution I will introduce GHD and discuss how, in combination with large deviation theory, it gives access to exact Euler scale (and beyond) correlation functions. Applications to XX and XY spin chain spin-spin correlation function and calculation of Renyi’s and entanglement entropy are presented.
_______________________________________________________________________________________________
Katja Koblas: Growth of Rényi entropies in interacting integrable models
After a quantum quench in a clean quantum many-body system Rényi entropies generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out-of-equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system's dynamics, and its characterisation is a key objective of current research. In the talk I will show that the slope of Rényi entropies can be determined by means of a spacetime duality transformation. This allows for an explicit exact formula for the slope of Rényi entropies in all integrable models treatable by thermodynamic Bethe ansatz. Interestingly, the formula can be understood in terms of a quasiparticle picture only in the von Neumann limit. The talk is based on Phys. Rev. X 12 031016 (2022).