Bonn Cologne seminar on Mathematics and Physics (BCoMP)
The seminar takes place online, biweekly on Mondays, at 4 pm (UTC+1)
Zoom link to seminar:
https://uni-koeln.zoom.us/j/92305906469?pwd=YjZBN0RjYzFYK1Exa09EaFFUeVpWZz09
SoSe 2022
- 11.07.2022
Sebastian Andres (U Manchester)
Heat kernel bounds for Liouville Brownian motion
In two-dimensional Liouville quantum gravity the main object of study is a random geometry on a domain D, which can be formally described by a Riemannian metric tensor of the form eγh(x) dx2 where h is a Gaussian free field on D and γ ∈ (0, 2) is a parameter. The natural diffusion process in this random geometry, called Liouville Brownian motion (LBM), can be constructed as a time-change of the planar Brownian motion on D. In this talk, we discuss sub-Gaussian heat kernel estimates for the LBM when γ = √8/3, which are sharp up to a polylogarithmic factor in the exponential. This value of γ is special because γ = √8/3- Liouville quantum gravity is equivalent to the Brownian map. This talk is based on a joint work in progress with Naotaka Kajino (Kyoto) and Jason Miller (Cambridge).
- 27.06.2022
Matthias Sperl (U Köln and DLR)
Aspects of Glassy Dynamics: Asymptotic Laws and Master Functions
Within the description of correlation functions in dense fluids, the dynamical laws arise from glass-transition singularities. The simplest such singularity yields a master function with a two-step decay. Higher-order singularities allow for logarithmic decay laws, while multiple singularities give rise to complex master functions.
While the scenarios with multiple singularities appear relevant for colloids, liquid water, and granular matter, the simple two-step scenario may be identified within simple models that allow for analytical calculations.
- 13.06.2022
Ood Shabtai (U Köln)
Pairs of spectral projections of spin operators
We study the semiclassical behavior of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, and contrast it with the behavior of the polynomial when evaluated on random pairs of projections.
- 23.05.2022
Yvan Velenik (U Genf)
Large-distance behavior of non-critical Ising correlations. Known results and open problems.
The large-distance asymptotic behavior of correlation functions (i.e., covariances of local random variables) in statistical mechanical systems have been investigated for more than a century, starting with the celebrated work of Ornstein and Zernike in 1914 and 1916. I'll review the state of the art in the context of the ferromagnetic Ising model on Z^{d}, focusing mainly on rigorous non-perturbative results valid at all non-critical temperatures.
- 9.05.2022
Muhittin Mungan (U Bonn)
Driving with maps: finding your way in a glassy energy landscape
Abstract: Understanding the response of a disordered solid to an externally imposed forcing or deformation is important in order to characterize the transitions between rigid and flowing states in a wide variety of soft matter systems, such as the yielding transition of an amorphous solid under shear. Such complex phenomena are already present in the "AQS" regime where thermal effects are negligible, the response to forcing is quasi-static, and the dynamics proceeds via triggering of mechanical instabilities. A key insight underlying my recent research has been the observation that the AQS conditions permit a rigorous representation of the driven dynamics in term of a directed state-transition graph. This AQS graph represents the response of the system to any deformation protocol, providing thereby a bird's-eye (or map-like) view of all the possible dynamics, which in turn is encoded in the topology of the AQS graph. At the same time, the AQS graph is a discrete random structure that lends itself to a mathematical treatment. In this talk I will describe the energy landscape of a sheared amorphous solid via AQS transition graphs, highlighting key features, such as hysteresis, and showing how various topological properties of this graph can be captured in terms of stochastic models.
WiSe 2021/22
- 24.01.2022
Nele Callebaut (U Köln)
Emergent gravity from 2D conformal field theory
Abstract: I will sketch two different ways in which gravity 'emerges' from a 2-dimensional CFT, and which type of mathematics are involved. Firstly, in AdS/CFT, 3D gravity gives a holographically dual description of the 2D CFT. The program of bulk reconstruction aims to identify how gravitationally interacting, 3D observables can be constructed from CFT observables. Secondly, TTbar theory describes how one can deform the 2D CFT with a particular irrelevant operator to effectively couple the CFT to 2D gravity.
- 10.01.2022
Peter Gracar (U Köln)
The many faces of the weight-dependent random connection model
Abstract: We take a look at several random geometric graphs (RGG) with increasing levels of complexity, starting from the classical Gilbert disc model with fixed radius and up to the weight-dependent random connection model. At each step, we discuss the heuristics of what the newly added complexity changes in the behavior of the models and how it affects the criticality of the largest connected component and the typical distance between two points of this component.
- 20.12.2021
Adrien Schertzer (U Bonn)
First passage percolation in the mean field limit.
Abstract: We will consider the oriented and unoriented first passage percolation, FPP for short, on the hypercube. In the oriented case, we prove that the extremal process converges to a Cox process with exponential intensity. This entails, in particular, that the first passage percolation (re-scaled and shifted) converges weakly to a random shift of the Gumbel distribution. In the second part of the talk, the assumption of directionality is dropped. We propose a constructive approach in order to identify the underlying strategies adopted by optimal paths to reach minimal values. Close to the origin, the optimal paths proceed in oriented fashion. The tension of the strand decreases however gradually, with the optimal paths allowing for more and more backsteps as it enters the core of the hypercube. Backsteps naturally increase the length of the strand, but also allow the optimal paths to connect reservoirs of energetically optimal edges which are otherwise unattainable in a fully directed regime. Somewhat surprisingly, the optimal paths manage to connect these reservoirs through approximate geodesics with respect to the Hamming metric: this is the key strategy which leads to an optimal energy/entropy balance. (Due to the inherent symmetry of the hypercube, a mirror picture naturally sets in around halfway). This approach yields the value of the unoriented first passage percolation as a corollary.
- 06.12.2021
Andreas Schadschneider (U Köln)
When order matters - Update dependence of steady states in discrete stochastic models
Abstract: In physics and applications, stochastic processes are often defined by local transition rules. These rules do not define the stochastic process uniquely. Different global updates exist which differ in the order in which the local rules are applied to the sites or particles. In the presentation the effect of the update type on the properties of the stationary state of the process is discussed mainly using the Asymmetric Simple Exlcusion Process (ASEP) and its generalizations as an example.
- 22.11.2021
no talk (Retreat Bad Honnef)
- 08.11.2021
Christian Brennecke (U Bonn)
Free Energy of the Quantum SK Model with Transverse Field
Abstract: In this talk I give an introduction to the quantum Sherrington-Kirkpatrick (SK) model with transverse field. I discuss basic results, including an approximation of the free energy in terms of a sequence of Parisi-like formulas of finite dimensional vector spin glasses, and present some open questions on the SK and other quantum spin glasses. The talk is based on joint work with A. Adhikari.
- 25.10.2021
Joachim Krug (U Köln)
Fitness landscapes and triangulations of the cube
Abstract: Darwinian evolution is driven by random mutations and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length $L$, an overview of possible evolutionary trajectories is provided by the oriented $L$-cube graph $\{0,1\}^L$ with nodes labeled by genotypes and edges directed toward the genotype with higher fitness. Peaks (sinks in the graphs) are important since a population can get stranded at a suboptimal peak. Some notion of curvature is necessary for a more complete analysis of the fitness landscape. For this purpose, the shape approach pioneered by Beerenwinkel, Pachter and Sturmfels (Statistica Sinica, 2007) uses triangulations of the genotope $[0,1]^L$ that are induced by the landscape. The main topic for this talk is the interplay between peak patterns and shapes. The talk is based on joint work with Kristina Crona and Malvika Srivastava.
SoSe 2021
- 19.07.21
Martin Zirnbauer (U Köln)
Color-Flavor Transformation Revisited
Abstract: The "color-flavor transformation", conceived as a kind of generalized Hubbard-Stratonovich transformation, is a variant of the Wegner-Efetov supersymmetry method for disordered electron systems. Tailored to quantum systems with disorder distributed according to the Haar measure of a compact Lie group of any classical type (A, B, C, or C), it has been applied to Dyson's Circular Ensembles, link-random network models, quantum chaotic graphs, disordered Floquet dynamics, and more. In this talk, I will review the method and discuss its limits of validity.
- 05.07.21
Gunter Schütz (U Bonn / FZ Juelich)
Integrability, supersymmetry and duality for vicious walkers with pair creation
Abstract: We study a system of independent random walkers in one dimension that annihilate immediately when two particles meet on the same site. In addition, pairs of particles are created randomly on neighbouring sites. For periodic boundary conditions, a duality with independent two-level systems is proved. The duality function, which arises from the free-fermion integrability of the model, is determinantal and can alternatively be represented in matrix product form. We use this duality to compute the exact current distribution. For reflecting boundaries the Markov generator commutes with the generators of a subalgebra of the universal enveloping algebra of the Lie superalgebra sl(1|1) and its deformations. The supersymmetry leads to a duality between an even and odd number of particles, respectively.
- 21.6.21
Andreas Schadschneider (U Köln, postponed)
postponed
- 07.06.21
Eveliina Peltola (U Bonn)
Towards a conformal field theory for Schramm-Loewner evolutions?
Abstract: For a number of critical lattice models in 2D statistical physics, it has been proven that scaling limits of interfaces (with suitable boundary conditions) are described by Schramm-Loewner evolution (SLE) curves. So-called partition functions of these SLEs (which also encode macroscopic crossing probabilities) can be regarded as specific correlation functions in the conformal field theory (CFT) associated to the lattice model in question. Although it is not clear how to define the latter mathematically, one can still make sense of many of the properties predicted for these CFTs. In particular, all of the expected CFT properties: conformal invariance, null-field equations, and fusion rules, are satisfied by the partition functions. One might then ask: Is it possible to go deeper and to construct the appropriate CFT fields as random distributions? Time permitting, I discuss some ideas to this direction.
- 17.05.21
Peter Mörters (U Köln)
Random partitions
Abstract: We study discrete stochastic processes that can be interpreted as evolving random partitions. Examples arise from random networks, population models with a family structure, or as the cycle structure of random permutations. The focus in the talk is on limit theorems for the relative size, the birth time and fitness of the largest partition set.
- 10.05.21
George Marinescu (U Köln)
Random polynomials and random holomorphic sections
Abstract: This talk focuses on the interplay between complex geometry and probability theory. More precisely, we will show how to use complex geometry and geometric analysis techniques in order to study several problems concerning local and global statistical properties of zeros of random polynomials.
- 26.04.21
Sebastian Diehl (U Köln)
Measurement induced phase transitions in monitored fermion chains
Abstract: Recently, a new class of phase transitions has been discovered, which result from a competition between deterministic Hamiltonian dynamics, and stochastic dynamics imprinted by local measurements. The transition surfaces in the dynamics of entanglement, and a transition from volume to area law was found in random circuit models. Here we establish a novel entanglement transition scenario between a regime of logarithmic entanglement growth, and a quantum Zeno regime obeying an area law, in continuously monitored fermion dynamics. Beyond the entanglement signatures, also correlation functions which are non-linear in the quantum state witness the transition. It interpolates between a gapless phase with algebraically decaying correlation functions, and a gapped one with exponential behavior. This motivates a statistical mechanics style approach to the problem, interpolating between the microscopic measurement dynamics and the macroscopic correlators. While the unread measurement dynamics heats up to infinity, the non-linear state evolution hosts degrees of freedom captured by a non-hermitean quantum Sine-Gordon model. This gives both a physical picture for the phase transition in terms of a depinning from the measurement operator eigenstates induced by unitary dynamics, and places it into the BKT universality class.
- 12.04.21
Corinna Kollath (U Bonn)
postponed
WiSe 2020/21
- 08.02.21
Lisa Hartung (U Mainz)
Entropic repulsion for the binary branching random walk
Abstract: Understanding entropic repulsion for the $2d$ discrete Gaussian free field is a major open problem. That is to understand the field when it is conditioned to be negative, We aim at getting a better understanding by taking a closer look at the corresponding question for the binary branching random walk (BRW). The latter has proven to be a good toy model for the 2d discrete Gaussian free field on the level of extreme values. We show that, under the conditioning, a uniformly chosen vertex (from an $n$-level BRW) will have height roughly $-m_{n-\log_2(n)} $, where $m_{n-\log_2(n)$ is the order of the maximum of a binary BRW with $n-\log_2(n)$ levels. This talk is based on joint work in progress with M. Fels.
- 25.01.21
Massimiliano Gubinelli (U Bonn)
Grassmann stochastic analysis and stochastic quantisation of Euclidean Fermions
Abstract: Abstract: This talk is about the extension of some probabilistic construction to the case of Grassmann valued random variables, i.e. random variables which anticommute. This require to set up the problem in the context of non-commutative probability. Moreover we study some simple stochastic differential equations for Grassmann variables and derive informations on their invariant states. Joint work with S. Albeverio, L. Borasi and F. de Vecchi. Based on the paper: Albeverio, Sergio, Luigi Borasi, Francesco C. De Vecchi, and Massimiliano Gubinelli. Grassmannian Stochastic Analysis and the Stochastic Quantization of Euclidean Fermions. arXiv:2004.09637.
- 11.01.21
Patrik Ferrari (U Bonn)
Local universality of the geodesic tree in last passage percolation
Abstract: We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. For several initial conditions, we prove that the local universality of the first order correction of the covariance when the two observation times are macroscopically small. We then show that also the geodesic tree in the same space-time windows is universal.
- 14.12.20
Margherita Disertori (U Bonn)
A supersymmetric transfer operator in 1D random band matrices.
Abstract: Transfer matrix approach is a powerful tool to study one dimensional or quasi 1d statistical mechanical models. Transfer operator kernels arising in the context of quantum diffusion and the supersymmetric approach display bosonic and fermionic components. For such kernels, the presence of fermion-boson symmetries allows to drastically simplify the problem. I will review the method and give some results for the case of random band matrices. This is joint work with Sasha Sodin and Martin Lohmann.
- 30.11.20
Alexander Drewitz (U Köln)
The (cable system) Gaussian free field: Towards some aspects of integrability and universality
Abstract: Critical exponents for percolation models have been computed for the model of Bernoulli percolation in special two-dimensional lattices in fundamental work by Smirnov and Werner, drawing on the powerful technique of conformal invariance and SLE. In this talk we outline an approach for determining various critical exponents for a percolation model on a wide range of transient graphs that is a functional of the (discrete) Gaussian free field, taking advantage of deep connections to the model of random interlacements. The results we obtain are universal in the sense that they do not depend on the local structure of the underlying graph. This is joint work with A. Prevost and P.-F. Rodriguez.
- 16.11.20
Joachim Krug (U Köln)
Inhomogeneous exclusion processes and the efficiency of translation
Abstract: Motivated by recent experiments on an antibiotic resistance gene, we investigate genetic interactions between synonymous mutations in the framework of inhomogeneous exclusion models of translation. We show that the range of possible interactions is markedly different depending on whether translation efficiency is assumed to be proportional to particle current or particle speed. In the first case every mutational effect, modeled as a change in one of the local jump rates, has a definite sign that is independent of the rate configuration, whereas in the second case the effect-sign can vary depending on the background. The first result is proved by mapping the exclusion process to last passage percolation, and the second result is demonstrated by approximately analysing configurations of multiple bottlenecks.
- 02.11.2020
Anton Bovier (U Bonn)
Brownian motion with social distancing
Abstract: We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term. This is joint work with Lisa Hartung.