English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Brownian force model (BFM) is the mean-field model for the avalanches ofan elastic interface slowly driven in a random medium. It describes thespatio-temporal statistics of the velocity field, and, to some extent isanalytically tractable. We extend our previous studies to obtain severalobservables for the BFM with short range elasticity, related to the local jumpsizes $S(x)$ and to the avalanche spatial extension in $d=1$, or the avalanchespan in $d>1$. In $d=1$ we consider both driving (i) by an imposed force (ii)by an imposed displacement "at the tip" and obtain in each case the meanspatial shape $\langle S(x) \rangle$ at fixed extension, or at fixed seed toedge distance. We find that near an edge $x_e$, $S(x) \simeq \sigma |x-x_e|^3$where $\sigma$ has a universal distribution that we obtain. We also obtain thespatiotemporal shape near the edge. In $d>1$ we obtain (i) the mean shape$\langle S(x_1,x_\perp) \rangle$ for a fixed span, which exhibits a non-trivialdependence in the transverse distance to the seed $x_\perp$ (ii) the mean shapearound a point which has not moved, $\langle S(x) \rangle_{S(0)=0}$, whichvanishes at the center as $|x|^{b_d}$ with non trivial exponents, $b_1=4$,$b_2=2 \sqrt{2}$ and $b_3=\frac{1}{2} (\sqrt{17}-1)$. We obtain the probabilitydistributions in any $d$ for the maximal radius of an avalanche and the minimaldistance of approach to a given point, as well as the probability of nothitting a cone in $d=2$. These results equivalently apply to the continuumlimit of some branching diffusions, as detailed in a companion paper.

More on the Brownian force model: avalanche shapes, tip driven, higher $d$