Week 09.02.2025 – 15.02.2025

This course will give a technical introduction to the black hole information paradox (BHIP). In the first lecture, we will begin with a review of quantum path integrals, both in Lorentzian and in Euclidean signature. We will review the Euclidean path integral connection with statistical mechanics and thermodynamics while also reviewing the derivation of the first law of thermodynamics in standard equilibrium statistical mechanics. We will then introduce the laws of black hole thermodynamics, and study them in particular examples. The second lecture will be devoted to the Unruh effect. We will study free quantum field theory in Rindler space, which, locally, is the spacetime observed by a uniformly accelerated observer. We will derive that this observer measures a temperature related to the observer proper acceleration. The third lecture will be devoted to classical and quantum information theory including notions of conditional probability, mutual information, and entropy inequalities, in settings with finite numbers of degrees of freedom. We will also introduce the Page curve and its significance. Finally, in the fourth lecture we will set up a toy model of the BHIP in Anti de Sitter space (AdS). Because AdS is believed to have a dual description as a conformal quantum field theory, we will use this duality to our advantage. We end with a broad discussion synthesizing what we have learned, and what is left to understand.

The dynamics of many-body systems, such as gases of particles or lattices of spins, often display, at large scales of space and time, a high degree of universality. Indeed, this dynamics is usually described by a few equations, those of hydrodynamics, representing
the flows of conserved currents such as those of particles and energy. This is because other "degrees of freedom" thermalise much more quickly, and the full dynamics projects onto that of conserved currents. In fact, surprisingly, even correlations between
local observables at large separations in time, and large-scale fluctuations, can be described by hydrodynamics. This is the object of various theories of hydrodynamic fluctuations, such as macroscopic fluctuation theory (for systems where diffusion dominates),
and its ballistic counterpart (for systems where persistent currents exist). I will introduce the main ideas behind such theories, restricting to systems in one dimension of space for simplicity. I will concentrate on perhaps the simplest and newest, ballistic
macroscopic fluctuation theory, taking simple examples such as the gas of classical hard rods (hard spheres, but in one dimension) - but many concepts are general.

I will talk about a general class of integrable models of interacting particles in inhomogeneous space, containing various types of inhomogeneous pushTASEPs and zero range processes, and how they are connected to determinantal point processes, random walks conditioned to never intersect and random tilings of the Aztec diamond with inhomogeneous weights. The integrability of these models comes from a natural generalisation of Toeplitz matrices which satisfy certain intertwining relations.

Encounter-based methods provide a general probabilistic framework for modelling adsorption on the surface or interior of a target. An adsorption event occurs when the contact time with the target exceeds a random threshold. If the probability distribution of the latter is an exponential function, then one recovers the Markovian example of adsorption at a constant rate, whereas a non-exponential distribution signifies non-Markovian adsorption. In the case of a partially adsorbing target surface (interior) the contact time is given by a Brownian functional known as the boundary local time (occupation time). In this talk we provide an overview of encounter-based methods. We begin by considering simple diffusive search processes. We then present several extensions of the theory such as search processes with stochastic resetting, active run-and-tumble particles, and diffusion across semipermeable membranes. Various applications to cell biology are also described.

In this talk, we discuss zero and uniqueness sets in spaces of entire functions. Our main tool involves locally convex plane curves. The results apply to discrete sets with random arguments.

Growth is a crucial feature of living systems, that sets them apart from most inanimate physical systems. I will discuss how statistical physics can shed light on the properties of growing living systems. Specifically, I will show how to use statistical physics to study growth of cell colonies and how their growth is coordinated with DNA replication. I will show how similar tool can be used to predict how epidemics
spread in complex networks. Finally, I will show examples of how growth generates exotic patterns in spatially extended biological systems