Bertrand Lacroix-A-Chez-Toine (King’s College London)

Superposition of random plane waves in high dimension as a random landscape  

Superpositions of random plane waves play important roles in the semi-classical description of quantum billiard as pointed out by Berry’s conjecture [1]. In this context, they can describe the eigenfunctions of the Laplacian operator at high energy. While their properties have been explored in depth in low spatial dimensions, we consider in this talk a large superposition of M ≫ 1 random plane waves in high dimension N ≫ 1 with M/N = α > 1. Here, we consider instead this object as a (random) energy landscape and, adding an isotropic harmonic confinement of strength μ, we characterise the ergodicity breaking in such a landscape. To characterise this property we consider two quantities: the quenched free energy and the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points of the energy landscape with the spatial dimension N. While similar high-dimensional random landscapes display topology trivialisation transition [2], whereby the complexity vanishes above some finite value of the confinement μ, the complexity vanishes only as μ → ∞ in this system [3]. One might thus expect that ergodicity is always broken at zero temperature in this model. This is confirmed and enriched by our quenched free-energy computations.

References

[1] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 2083 (1977).

[2] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[3] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, J. Math. Phys. 63 (9), 093301 (2022)

Jim Sethna (Cornell University)

Using Universal Scaling Functions

Half a century ago, Ken Wilson and Leo Kadanoff introduced the renormalization-group framework for understanding systems with emergent, fractal scale invariance. For five decades, statistical physicists have applied these techniques to equilibrium phase transitions, avalanche models, glasses and disordered systems, the onset of chaos, plastic flow in crystals, surface morphologies, fracture, … But these tools have not made a substantial impact in engineering or biology.

Even now we do not have a clear understanding of the singularity at the critical point for even the traditional equilibrium phase transitions – we know the critical exponents to high precision, but are missing complete understanding of the universal scaling functions. Also, we have not developed the tools to extend our predictions of the asymptotic behavior to a systematic approximation of the phase diagrams – analytic corrections to scaling. Sethna will begin with an introduction to how normal form theory from dynamical systems can determine the arguments of scaling functions where logarithms and exponentials intrude on the traditional power laws – the 4D Ising model, the 2D random-field Ising model, and jamming and the onset of amorphous rigidity in 2D. He will then use the 2D Ising model to show how adding analytic corrections to the critical point singularity can lead to high-precision descriptions of the entire phase diagram, and present Jaron Kent-Dobias’s derivation of a universal scaling function for the free energy in an external field, accurate to seven digits.

Xiangyu Cao (ENS Paris)

Clusters in branching processes

Branching Brownian motion and its long-range generalisations are simple models of a variety of catastrophic phenomena, such as epidemic spreading and avalanches. Despite the simplicity of these models, they generate interesting geometric structures, in particular, clusters. I will discuss a simple method to count clusters and characterise their spatial separation, which turn out to be governed by nontrivial critical exponents. As an application, I will test the theory against real-world data generated by a famous recent epidemic outbreak.

Charlie Duclut (Institut Curie)

Renormalization group approach to the collective dynamics of chemotactic cells

Understanding how living systems self-organize into complex structures is one of the major challenges of modern physics. A generic mechanism that drives such organization is interaction among the individual elements — which may represent cells, bacteria, or even enzymes — via chemical signals. After deriving a minimal microscopic model for a single chemotactic particle, I will present a coarse-grained model to describe an assembly of such particles. I will be interested both in the case where cell number is constant, and in the case where cell can grow and divide. The consequences of breaking this conservation law on the scaling properties of this model will be discussed using a dynamical renormalization group approach.