Bonn Cologne seminar on Mathematics and Physics (BCoMP)

Zoom link to seminar:

WiSe 2021/22

  • 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)

  • 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)

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.