# Internships (Stages) + PhD Proposals (Thèses)

Applications to write a PhD thesis, a master thesis on the M1 or M2 level, or to do bibliographic research under my guidance are welcome at all times. I am happy to discuss possible projects centered around my research interestst or your interest with you. This is best discussed in person. Please contact me if you are interested. Three current topics are listed below.

### Topic 1: Unzipping of RNA (M1 or M2 internship)

One of the most exciting experimental developments at the frontier between physics and biology is our possiblity to attach the ends of an RNA or DNA molecule to beads, trap the beads in an optical tweezer, pull on the beads, and measure the ensuing forces as a function of bead-position (see figure). This force in the pico-Newton range contains a wealth of information on the biological system. It is also an exciting testing ground for theoretical concepts. In this internship, we propose to study the effects of thermal fluctuations onto the sawtooth like unzipping curves. Are these thermal effects negligible, or do they even change the large-scale physics?

### Topic 2: Extreme-Value Statistics for Fractional Brownian Motion (M1 or M2 internship, PhD project)

Brownian motion is a stochastic process which is Gaussian, scale invariant, translationally invariant, and Markovian, i.e. has independent increments. Apart from simple expectations, one is particularly interested in extremal properties: Does the process exceed a curtain threshold, and if yes, when? How long is the process positive? This is important to correctly model the necessary capacity for dams, estimate the time one has to heat, etc. While these questions can usually be answered for Brownian motion, which is Markovian (a property essential in the solution), many processes in nature are not Markovian, i.e. have memory. Fractional Brownian motion (see figure) is such a generalization, indexed by the Hurst exponent H, and reducing for H = 1/2 to Brownian motion. For H < 1/2, the process is anticorrelated and rougher (red curves in the figure), while for H > 1/2 it is positively correlated. We have recently constructed a systematic perturbative expansion around H = 1/2. There are still unexplored applications, e.g. in the 2-sided exit problem, relevant e.g. for translocation of a polymer through a nanopore. We wish to use integrable systems, which provide at least for H = 1/4 another exactly solvable model. (PhD level project).

### Topic 3: Non-standard field theories (PhD project)

Field theory is the most powerful method developed by the human mind to understand scale invariant systems. Examples in case are (i) the particle accelerator at CERN, (ii) critical phenomena as the the liquid-vapor transition or the ferro-para transition in magnets, (iii) disordered systems. Field theories for the latter have some unusual features, as non-polynomial and even non-analytic fixed points, which render its analysis more involved. Recent research [1] has shown that behind these apparent complications are hidden simpler analytical structures. It would be interesting to exploit these more broadly. Other questions handled in this proposal are avalanhes, especially their spatial shape, recently solved in the MF limit [2].
[1] K.J. Wiese and A.A. Fedorenko, Field theories for loop-erased random walks, (2018), arXiv:1802.08830
[2] Z. Zhu and K.J. Wiese, The spatial shape of avalanches, Phys. Rev. E 96 (2017) 062116, arXiv:1708.01078.

Copyright (C) by Kay Wiese. Last edited March 20, 2018.