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A Large-Deviation Study

Random Walks

- position $\vec{x}(t)$ is sum of random Gaussian steps $\vec\delta_i$ $$\vec x(t) = \sum_{i=1}^t \vec\delta_i, \quad t \le T$$
- same asymptotics when on a lattice
- scales as $r \propto T^{\nu}, \nu=1/2$
- model for diffusion/Brownian motion
- simple model for animal movement

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$T=500$

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Convex Hulls

- smallest convex polytope around points $$\{x(t) | t \in \{0, .., T\}\}$$

Divide and Conquer:

Quickhull

[1] C. B. Barber, D. P. Dobkin, and H. Huhdanpaa,
ACM TOMS **22**, 469 (1996)

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$T=500$

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- characterizes a random walk
- $\partial V$ surface
- $V$ volume
- e.g. home ranges of animals in $d=2$
- mean values known for $T\to\infty$
^{[2]}$$\left< V \right> / T^{d/2} = \left( \frac{\pi}{2} \right)^{d/2} \Gamma \left( \frac{d}{2} + 1 \right)^{-2}$$ - we will look at their
**full distribution**

[2] R. Eldan, Electron. J. Probab. **19**, 45 (2014)

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Sampling

- Simple sampling works only for high $P$
- What about the $P \ll 10^{-100}$ tails?
- (modified
^{[4]}) Wang Landau sampling^{[3]} - accept with Metropolis probabilty

- iteratively adapt $g$ $\to$ flat histogram

[3] F. Wang and D. P. Landau, Phys. Rev. Lett. **86**, 2050 (2001)

[4] R. E. Belardinelli and V. D. Pereyra, Phys. Rev. E**75**, 046701 (2007)

[4] R. E. Belardinelli and V. D. Pereyra, Phys. Rev. E

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- classical
^{[3]}:- $\ln g_{i+1} = \ln g_{i} + f$
- $f \mapsto f/2$ after every histogram reset

- modifying $g$ $\to$ no detailed balance
- to reduce systematic error, different schedule
^{[4]}($t$ is Monte Carlo time) - $\ln g_{i+1} = \ln g_{i} + t^{-1}$

[3] F. Wang and D. P. Landau, Phys. Rev. Lett. **86**, 2050 (2001)

[4] R. E. Belardinelli and V. D. Pereyra, Phys. Rev. E**75**, 046701 (2007)

[4] R. E. Belardinelli and V. D. Pereyra, Phys. Rev. E

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Results

$d=4$

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Observables scale as $T^{d/2}$ with their effective dimension.

Here $d=4$ dimensional hypervolume $V$.

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$$\widetilde{S}^{(d-1)/d} \mathrm{e}^{-b\widetilde{S}^{2/d}}$$
from the distribution of the $d=1$ span

G. Claussen, A. K. Hartmann, and S. N. Majumdar, Phys. Rev. E **91**, 052104 (2015)

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- The large deviation properties of the $d=1$ span of random walks seem to be easily extended to larger $d$.

- Multiple Walks
- Self-Interacting Walks (SAW, LERW, ...) Poster session DY 60.1 on Thursday

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