# Theory Club

## Explanatory note

This is a roughly weekly series of didactical blackboard talks focussing on some theoretical aspect of Condensed Matter, Biological, and Statistical Physics. The range of topics is sufficiently loosely defined that core interests within SUPA's CMMP theme, the agent-based modelling of complex systems relevant to the NANIA project, or whatever else takes our fancy are all accommodated.

Actually, the best way to find out what we talk about is to look at the list of titles below.

If you're wanting to receive mailings about these talks, request a topic for discussion, or - even better - offer to lead the discussion yourself, please email us Juraj Szavits-Nossan, Peter Mottishaw.

## Week beginning 14 September 2014

Wednesday 17 Sep 14 - 11:30am - **2511**

Correlated Extreme Values in Branching Brownian Motion

Kabir Ramola (LPTMS, France)

We investigate one dimensional branching Brownian motion (BBM) in which at each time step particles either diffuse (with diffusion constant D), die (with rate d), or split into two particles (with rate b). When the birth rate exceeds the death rate (b > d), there is an exponential proliferation of particles and the process is explosive. When b < d, the process eventually dies. At the critical point (b = d) this system is characterized by a fluctuating number of particles which individually behave diffusively. Quite remarkably, although the individual positions of these particles have a non-trivial finite time behaviour, the average distances between successive particles (the gaps) become stationary at large times, implying strong correlations between particles. We compute the probability distribution functions (PDFs) of these gaps, by conditioning the system to have a fixed number of particles at a given time t. At large times we show that these PDFs are characterized by a power law tail ~1/g^3 (for large gaps g) at the critical point and ~exp(- g/c) otherwise, with a correlation length c~(D/|b - d|)^(1/2). We discuss the emergence of these two length scales, the dominant overall length scale of the individual positions, and the sub-dominant gap length scale in this system. We also extend our study to the spatial extent of this process (the distance between the rightmost and leftmost visited sites). We derive exact results for the PDF of this spatial extent for the cases b <= d where the two extreme points are strongly correlated. Once again we find an emergent power law at the critical point with a correlation length ~(D//|b - d|)^(1/2) away from criticality. Direct Monte Carlo simulations confirm our predictions.

## Upcoming meetings

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