The Fröhlich polaron models a charged quantum particle interactiong with a polar cystal. Since the moving particle has to drag along a ‘cloud’ of polarization, it appears heavier than it would be without the interaction. An old conjecture of Landau and Pekar states that this so-called effective mass scales as the fourth power of the […]
It is known through classical works of Kac, Salem, Zygmund, Erdös and Gal that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem or a law of the iterated logarithm. We present some recent results on their large deviation behavior, which show that on this […]
In this talk – based on a joint work with J. Lampart, N. Leopold, and D. Mitrouskas – I will talk about the mean field limit of the renormalized Nelson model, in which a large number of bosonic particles is weakly coupled with a large number of coherent excitations of the scalar field. We prove […]
The meeting of the junior members of TRR 352 took place from December 13-15, 2023, at the Haus der bayerischen Landwirtschaft in Herrsching/Ammersee. program titles & abstracts Speakers: Alberto Brollo (TUM) Florian Haberberger (LMU Munich) Paul Hege (Tübingen University) Viet Hoang (Tübingen University) Zahra Khanian (TUM) Peter Madsen (LMU Munich) Larry Read (LMU Munich) Diane Saint Aubin (Zurich […]
We consider a random walk in a truncated cone K_N , which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic […]
In this talk, we investigate random convex interfaces which are generated as convex hulls of random point sets. We are interested in their asymptotic behavior when the size of the input goes to infinity. In a first part, we mainly identify average and maximal fluctuations in the radial and longitudinal directions through precise convergence results. […]
Interacting Bose gas at zero temperature is often described by the Bogoliubov approximation. It involves quasiparticles, called phonons, with a rather curious dispersion relation responsible for superfluidity. The Fermi Golden Rule predicts that the lifetime of phonons is proportional to the 5th inverse power of momentum. This was first computed by Beliaev and goes under […]
In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation […]
The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G. The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed […]
From October 11 to 15, 2023, the kick-off meeting of our CRC took place in Farchant near Garmisch. Around 60 participants came together to spend a few days in the beautiful surroundings of the Bavarian Alps. The project leaders had the opportunity to present their projects and report on current progress and ideas. On Friday […]
The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, […]
We introduce a currently hot topic in probability theory, the theory of scaling limits for random fields of gradients in all dimensions. The random fields are a class of model systems arising in the studies of random interfaces, random geometry, Euclidean field theory, the theory of regularity structures, and elasticity theory. After explaining how non-convex […]
I will present a new approach to finding the asymptotic states of Nonlinear Wave Equations with general initial data. In particular, we show for a large class of equations, that all asymptotic states are linear combinations of free wave, localized parts (solitons, breathers..) and a possibility of self-similar solutions as well in some cases. These […]
The homogeneous electron gas (jellium) where electrons interact with each other and with a positive background charge is one of the simplest model system in condensed matter physics. Still, the precise determination of the zero temperature phase diagram remains challenging. In the talk I will review some recent progress from a computational perspective concerning the […]
We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose-Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables. We prove that in the limit N to infinity, […]
In this talk, we describe the kinetic equation for the Bogoliubov excitations of the Bose-Einstein Condensate. We find three collisional processes: One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves. This is a joint work with […]
Propagation and generation of “chaos” is an important ingredient in rigorous control of applicability of kinetic theory, in general. Chaos can here be understood as sufficient statistical independence of random variables related to the “kinetic” obser- vables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution […]
We consider Markovian open quantum dynamics (MOQD) in continuous space. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution […]
In this talk I will describe the vacuum sector of the Weinberg-Salam (WS) model of electroweak forces. In the vacuum sector the WS model yields the U(2)-Yang-Mills-Higgs equations. We show that at large constant magnetic fields the translational symmetry of the equations is broken spontaneously. Namely, there are solutions, which in the plane orthogonal to […]
We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying […]
Unraveling the origin of unconventional superconductivity is one of the driving forces behind quantum simulations with Fermions in optical lattices. In these strongly correlated materials, the necessary pairing of charge carriers is often assumed to be related to the interplay of antiferromagnetic correlations and dopant motion. Despite impressive recent progress in the numerical treatment of […]
We consider the infrared problem in translation-invariant Nelson-type models describing a single quantum mechanical particle linearly coupled to a field of scalar bosons at fixed total momentum. Physical examples include the non- and semi-relativistic Nelson models. If the bosons are massless, then the model is infrared divergent and the infimum of the spectrum is not […]
The correspondence principle, as stated by Niels Bohr in 1923, is at the root of the traditional results in semi-classical analysis. It offers a natural insight into the world of semi- classical pseudodifferential operators, Egorov Theorem, coherent states, Wigner measures, etc… The aim of this talk will be to present this general setting and explain […]
Schrödinger’s equation is a beautiful piece of mathematics. It f its on just one line and is supposed to accurately describe the behavior of most atoms and molecules of our world. But it is essentially impossible to simulate accurately, due to its very high dimensionality. In this talk I will explain how physicists and chemists […]
Recent progress in the development of quantum technologies has enabled the direct investigation of dynamics of increasingly complex quantum many-body systems. This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage. Here we present classical algorithms for approximating […]
A mathematical understanding of the mechanism of metallic ferromagnetism still needs to be completed. In this talk, the following three fundamental theorems on metallic ferromagnetism will be first outlined: the Marshall-Lieb-Mattis theorem, the Lieb theorem, and the stability theorem of Lieb ferrimagnetism. Next, I will outline a mathematical framework within which these theorems can be […]
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed […]
Although many-body quantum simulations have greatly benefited from high-perfor- mance computing facilities, large molecular systems continue to pose formidable challenges. Mixed quantum-classical models, such as Born-Oppenheimer molecular dynamics or Ehren- fest dynamics, have been proposed to overcome the computational costs of fully quantum ap- proaches. However, current mixed quantum-classical models typically suffer from long- standing […]
The Fröhlich polaron models a charged quantum particle interactiong with a polar cystal. Since the moving particle has to drag along a ‘cloud’ of polarization, it appears heavier than it would be without the interaction. An old conjecture of Landau and Pekar states that this so-called effective mass scales as the fourth power of the […]
It is known through classical works of Kac, Salem, Zygmund, Erdös and Gal that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem or a law of the iterated logarithm. We present some recent results on their large deviation behavior, which show that on this […]
In this talk – based on a joint work with J. Lampart, N. Leopold, and D. Mitrouskas – I will talk about the mean field limit of the renormalized Nelson model, in which a large number of bosonic particles is weakly coupled with a large number of coherent excitations of the scalar field. We prove […]
The meeting of the junior members of TRR 352 took place from December 13-15, 2023, at the Haus der bayerischen Landwirtschaft in Herrsching/Ammersee. program titles & abstracts Speakers: Alberto Brollo (TUM) Florian Haberberger (LMU Munich) Paul Hege (Tübingen University) Viet Hoang (Tübingen University) Zahra Khanian (TUM) Peter Madsen (LMU Munich) Larry Read (LMU Munich) Diane Saint Aubin (Zurich […]
We consider a random walk in a truncated cone K_N , which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic […]
In this talk, we investigate random convex interfaces which are generated as convex hulls of random point sets. We are interested in their asymptotic behavior when the size of the input goes to infinity. In a first part, we mainly identify average and maximal fluctuations in the radial and longitudinal directions through precise convergence results. […]
Interacting Bose gas at zero temperature is often described by the Bogoliubov approximation. It involves quasiparticles, called phonons, with a rather curious dispersion relation responsible for superfluidity. The Fermi Golden Rule predicts that the lifetime of phonons is proportional to the 5th inverse power of momentum. This was first computed by Beliaev and goes under […]
In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation […]
The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G. The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed […]
From October 11 to 15, 2023, the kick-off meeting of our CRC took place in Farchant near Garmisch. Around 60 participants came together to spend a few days in the beautiful surroundings of the Bavarian Alps. The project leaders had the opportunity to present their projects and report on current progress and ideas. On Friday […]
The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, […]
We introduce a currently hot topic in probability theory, the theory of scaling limits for random fields of gradients in all dimensions. The random fields are a class of model systems arising in the studies of random interfaces, random geometry, Euclidean field theory, the theory of regularity structures, and elasticity theory. After explaining how non-convex […]
I will present a new approach to finding the asymptotic states of Nonlinear Wave Equations with general initial data. In particular, we show for a large class of equations, that all asymptotic states are linear combinations of free wave, localized parts (solitons, breathers..) and a possibility of self-similar solutions as well in some cases. These […]
The homogeneous electron gas (jellium) where electrons interact with each other and with a positive background charge is one of the simplest model system in condensed matter physics. Still, the precise determination of the zero temperature phase diagram remains challenging. In the talk I will review some recent progress from a computational perspective concerning the […]
We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose-Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables. We prove that in the limit N to infinity, […]
In this talk, we describe the kinetic equation for the Bogoliubov excitations of the Bose-Einstein Condensate. We find three collisional processes: One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves. This is a joint work with […]
Propagation and generation of “chaos” is an important ingredient in rigorous control of applicability of kinetic theory, in general. Chaos can here be understood as sufficient statistical independence of random variables related to the “kinetic” obser- vables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution […]
We consider Markovian open quantum dynamics (MOQD) in continuous space. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution […]
In this talk I will describe the vacuum sector of the Weinberg-Salam (WS) model of electroweak forces. In the vacuum sector the WS model yields the U(2)-Yang-Mills-Higgs equations. We show that at large constant magnetic fields the translational symmetry of the equations is broken spontaneously. Namely, there are solutions, which in the plane orthogonal to […]
We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying […]
Unraveling the origin of unconventional superconductivity is one of the driving forces behind quantum simulations with Fermions in optical lattices. In these strongly correlated materials, the necessary pairing of charge carriers is often assumed to be related to the interplay of antiferromagnetic correlations and dopant motion. Despite impressive recent progress in the numerical treatment of […]
We consider the infrared problem in translation-invariant Nelson-type models describing a single quantum mechanical particle linearly coupled to a field of scalar bosons at fixed total momentum. Physical examples include the non- and semi-relativistic Nelson models. If the bosons are massless, then the model is infrared divergent and the infimum of the spectrum is not […]
The correspondence principle, as stated by Niels Bohr in 1923, is at the root of the traditional results in semi-classical analysis. It offers a natural insight into the world of semi- classical pseudodifferential operators, Egorov Theorem, coherent states, Wigner measures, etc… The aim of this talk will be to present this general setting and explain […]
Schrödinger’s equation is a beautiful piece of mathematics. It f its on just one line and is supposed to accurately describe the behavior of most atoms and molecules of our world. But it is essentially impossible to simulate accurately, due to its very high dimensionality. In this talk I will explain how physicists and chemists […]
Recent progress in the development of quantum technologies has enabled the direct investigation of dynamics of increasingly complex quantum many-body systems. This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage. Here we present classical algorithms for approximating […]
A mathematical understanding of the mechanism of metallic ferromagnetism still needs to be completed. In this talk, the following three fundamental theorems on metallic ferromagnetism will be first outlined: the Marshall-Lieb-Mattis theorem, the Lieb theorem, and the stability theorem of Lieb ferrimagnetism. Next, I will outline a mathematical framework within which these theorems can be […]
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed […]
