arXiv subjects

Julia Überbacher

Publications and source records attributed to Julia Überbacher.

3 recordsLinked to original sources

arXiv2026

Limit shape, avalanches, and stabilisation of abelian sandpiles on comb lattices

We study two aspects of the abelian sandpile model on comb lattices. First, we prove that the infinite-volume limit of the stationary measures is supported on the saturated configuration. We then investigate the shape of avalanches induced by adding a particle at the origin in finite boxes of size $(2n+1)\times(2n+1)$ around the origin. In the stationary distribution on these finite boxes, we show that avalanches reach the boundary along the vertical teeth with probability tending to $1$, while the horizontal spread is of order $\sqrt{n}$. Finally, we establish that the single-source limit shape for abelian sandpiles on the comb lattice agrees with the corresponding limit shapes for the divisible sandpile, internal diffusion-limited aggregation (IDLA), and rotor-router aggregation models. This establishes limit shape universality on the comb lattice.

math.PR↗

arXiv2026

On the structure of the sandpile identity element on Sierpinski gasket graphs

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpinski gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.

math.CO↗

arXiv2026

Limit shape of single-source stochastic sandpiles with $p$-topplings on $\mathbb{Z}$

We investigate the limit shape of the single-source model for stochastic sandpiles on the integer line subject to $p$--topplings. In this model, an initial configuration of $n\in\mathbb{N}$ particles is placed at the origin and stabilized according to a random toppling rule depending on $p\in (0,1)$: an unstable vertex sends exactly one particle to its left neighbor with probability $p$, and independently sends exactly one particle to its right neighbor with probability $p$. We prove that as $n \to \infty$, the macroscopic limit shape of the final stable configuration is a symmetric interval around the origin. Furthermore, by analyzing the center of mass martingale, we establish a central limit theorem for the boundary fluctuations, showing that after proper rescaling, they converge to a Gaussian distribution.

math.PR↗