arXiv subjects

Dominik Schmid

Publications and source records attributed to Dominik Schmid.

At least 19 recordsLinked to original sources

arXiv2026

Cluster-Cluster model in $\mathbb{Z}^d$

We consider a stochastic process on $\mathbb{Z}^d$ for $d \geq 1$. Given a translation invariant and ergodic starting configuration of finite clusters, each cluster $C$ performs a continuous time simple random walk with rate $|C|^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In all dimensions, we show that if $\alpha\ge 0$, there is almost surely no spontaneous creation of an infinite cluster within finite time. Moreover, for any $\alpha\le-1-2/d$ there is a finite-time blowup almost surely. In the regime $\alpha\in(-1,0)$ we show that the behavior greatly depends on the initial configuration. In addition, in dimension one, we establish the exact phase diagram.

math.PR↗

arXiv2026

Periodic directed landscape

We construct the periodic directed landscape, which is the conjectured scaling limit for periodic models in the Kardar-Parisi-Zhang universality class. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape. Moreover, we confirm conjectures by Baik, Liao and Liu on the local structure of the periodic KPZ fixed point, and establish the convergence of periodic ASEPs to periodic KPZ fixed points, coupled according to the same periodic directed landscape. Our main tool to construct the periodic directed landscape (and prove convergence to it) is a technique for gluing full-space directed landscapes (and their prelimits), which is of independent interest.

math.PR↗

arXiv2025

Cutoff for generalised Bernoulli-Laplace urn models

We introduce a multi-colour multi-urn generalisation of the Bernoulli-Laplace urn model, consisting of $d$ urns, $m$ colours, and $dmn$ balls, with $dn$ balls of each colour and $mn$ balls in each urn. At each step, one ball is drawn uniformly at random from each urn, and the chosen balls are redistributed among the urns based on a permutation drawn from a distribution $\mu$ on the symmetric group $S_d$. We study the mixing time of this Markov chain for fixed $m$, $d$, and $\mu$, as $n \rightarrow \infty$. We show that there is cutoff whenever the chain on $[d]$ corresponding to the evolution of a single ball is irreducible, and that the same holds for a labeled version of the model. As an application, we also obtain partial results on cutoff for a card shuffling version of the model in which the cards are labeled and their ordering within each stack matters.

math.PR↗

arXiv2025

Optimally building spanning graphs in semirandom graph processes

The semirandom graph process constructs a graph $G$ in a series of rounds, starting with the empty graph on $n$ vertices. In each round, a player is offered a vertex $v$ chosen uniformly at random, and chooses an edge on $v$ to add to $G$. The player's aim is to make $G$ satisfy some property as quickly as possible. Our interest is in the property that $G$ contain a given $n$-vertex graph $H$ with maximum degree $\Delta$. In 2021, Ben-Eliezer, Gishboliner, Hefetz and Krivelevich showed that there is a semirandom strategy that achieves this, with probability tending to 1 as $n$ tends to infinity, in $(1 + o_\Delta(1)) \frac{3 \Delta n}{2}$ rounds, where $o_\Delta(1)$ is a function that tends to $0$ as $\Delta$ tends to infinity. We improve this to $(1 + o_\Delta(1)) \frac{\Delta n}{2}$, which can be seen to be asymptotically optimal in $\Delta$. We show the same result for a variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on $n$ vertices, and chooses one edge to keep.

math.CO↗

arXiv2025

MeshFleet: Filtered and Annotated 3D Vehicle Dataset for Domain Specific Generative Modeling

Generative models have recently made remarkable progress in the field of 3D objects. However, their practical application in fields like engineering remains limited since they fail to deliver the accuracy, quality, and controllability needed for domain-specific tasks. Fine-tuning large generative models is a promising perspective for making these models available in these fields. Creating high-quality, domain-specific 3D datasets is crucial for fine-tuning large generative models, yet the data filtering and annotation process remains a significant bottleneck. We present MeshFleet, a filtered and annotated 3D vehicle dataset extracted from Objaverse-XL, the most extensive publicly available collection of 3D objects. Our approach proposes a pipeline for automated data filtering based on a quality classifier. This classifier is trained on a manually labeled subset of Objaverse, incorporating DINOv2 and SigLIP embeddings, refined through caption-based analysis and uncertainty estimation. We demonstrate the efficacy of our filtering method through a comparative analysis against caption and image aesthetic score-based techniques and fine-tuning experiments with SV3D, highlighting the importance of targeted data selection for domain-specific 3D generative modeling.

cs.CV↗

arXiv2025

Biased random walk on the critical curve of dynamical percolation

We study biased random walks on dynamical percolation in $\mathbb{Z}^d$, which were recently introduced by Andres et al. We provide a second order expansion for the asymptotic speed and show for $d \ge 2$ that the speed of the biased random walk on the critical curve is eventually monotone increasing. Our methods are based on studying the environment seen from the walker as well as a combination of ergodicity and several couplings arguments.

math.PR↗

arXiv2025

Mixing times for the open ASEP at the triple point

We consider mixing times for the open asymmetric simple exclusion process (ASEP) at the triple point. We show that the mixing time of the open ASEP on a segment of length $N$ for bias parameter $q$ is of order $N^{3/2+\kappa}$ if $1-q \asymp N^{-\kappa}$ for some $\kappa \in [0,\frac{1}{2})$, and the same result with poly-logarithmic corrections for $\kappa=\frac{1}{2}$. Our proof combines a fine analysis of the current of the open ASEP, moderate deviations of second class particles, the censoring inequality, and various couplings and multi-species extensions of the ASEP. Moreover, we establish a comparison between moderate deviations for the current of the open ASEP and the ASEP on the integers, as well as bounds on mixing times for the open ASEP in the weakly high density phase, which are of independent interest.

math.PR↗

arXiv2024

Limit Profile for the Bernoulli--Laplace Urn

We analyse the convergence to equilibrium of the Bernoulli--Laplace urn model: initially, one urn contains $k$ red balls and a second $n-k$ blue balls; in each step, a pair of balls is chosen uniform and their locations are switched. Cutoff is known to occur at $\tfrac12 n \log \min\{k, \sqrt n\}$ with window order $n$ whenever $1 \ll k \le \tfrac12 n$. We refine this by determining the limit profile: a function $\Phi$ such that \[ d_\mathsf{TV}\bigl( \tfrac12 n \log \min\{k, \sqrt n\} + \theta n \bigr) \to \Phi(\theta) \quad\text{as}\quad n \to \infty \quad\text{for all}\quad \theta \in \mathbb R. \] Our main technical contribution, of independent interest, approximates a rescaled chain by a diffusion on $\mathbb R$ when $k \gg \sqrt n$, and uses its explicit law as a Gaussian process.

math.PR↗

arXiv2024

The open ASEP with light particles

We consider the stationary measure of the open asymmetric simple exclusion process (ASEP) with light particles. We prove several results on the asymptotic locations of the light particles under the stationary measure. Moreover, in the fan region of the high and low density phases, we study the mixing times for open ASEP with light particles. Our approach involves the matrix product ansatz, Askey-Wilson signed measures, moderate deviation results for second class particles, as well as various coupling arguments. Along the way we obtain the particle densities at single sites for the standard open ASEP without light particles, which are of independent interest.

math.PR↗

arXiv2024

Bootstrap percolation on the high-dimensional Hamming graph

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get infected in subsequent rounds if they have at least $r$ infected neighbours. A graph $G$ \emph{percolates} if every vertex becomes eventually infected. A central problem in this process is to determine the critical probability $p_c(G,r)$, at which the probability that $G$ percolates passes through one half. In this paper, we study random $2$-neighbour bootstrap percolation on the $n$-dimensional Hamming graph $\square_{i=1}^n K_k$, which is the graph obtained by taking the Cartesian product of $n$ copies of the complete graph $K_k$ on $k$ vertices. We extend a result of Balogh and Bollob\'{a}s [Bootstrap percolation on the hypercube, Probab. Theory Related Fields. 134 (2006), no. 4, 624-648. MR2214907] about the asymptotic value of the critical probability $p_c(Q^n,2)$ for random $2$-neighbour bootstrap percolation on the $n$-dimensional hypercube $Q^n=\square_{i=1}^n K_2$ to the $n$-dimensional Hamming graph $\square_{i=1}^n K_k$, determining the asymptotic value of $p_c\left(\square_{i=1}^n K_k,2\right)$, up to multiplicative constants (when $n \rightarrow \infty$), for arbitrary $k \in \mathbb N$ satisfying $2 \leq k\leq 2^{\sqrt{n}}$.

math.CO↗

arXiv2023

Catching a robber on a random $k$-uniform hypergraph

The game of \emph{Cops and Robber} is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The \emph{cop number} of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an $n$-vertex connected graph is $O(\sqrt{n})$. In 2016, Pra{\l}at and Wormald [Meyniel's conjecture holds for random graphs, Random Structures Algorithms. 48 (2016), no. 2, 396-421. MR3449604] showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreoever, {\L}uczak and Pra{\l}at [Chasing robbers on random graphs: Zigzag theorem, Random Structures Algorithms. 37 (2010), no. 4, 516-524. MR2760362] showed that on a $\log$-scale the cop number demonstrates a surprising \emph{zigzag} behaviour in dense regimes of the binomial random graph $G(n,p)$. In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the $k$-uniform binomial random hypergraph $G^k(n,p)$ is $O\left(\sqrt{\frac{n}{k}}\, \log n \right)$ for a broad range of parameters $p$ and $k$ and that on a $\log$-scale our upper bound on the cop number arises as the minimum of \emph{two} complementary zigzag curves, as opposed to the case of $G(n,p)$. Furthermore, we conjecture that the cop number of a connected $k$-uniform hypergraph on $n$ vertices is $O\left(\sqrt{\frac{n}{k}}\,\right)$.

math.CO↗

arXiv2023

Limit profile for the ASEP with one open boundary

We study the speed of convergence to equilibrium for the asymmetric simple exclusion process (ASEP) on a finite interval with one open boundary. We provide sharp estimates on the total-variation distance from equilibrium and verify that the limit profile undergoes a phase transition from a Gaussian to a KPZ profile.

math.PR↗

arXiv2023

Approximating the stationary distribution of the ASEP with open boundaries

We investigate the stationary distribution of asymmetric and weakly asymmetric simple exclusion processes with open boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow with the length of the underlying segment. Depending on the boundary parameters of the exclusion process, we provide conditions such that the stationary distribution projected onto a subinterval is close in total variation distance to a product measure.

math.PR↗

arXiv2023

The $S_k$ shuffle block dynamics

We introduce and analyze the $S_k$ shuffle on $N$ cards, a natural generalization of the celebrated random adjacent transposition shuffle. In the $S_k$ shuffle, we choose uniformly at random a block of $k$ consecutive cards, and shuffle these cards according to a permutation chosen uniformly at random from the symmetric group on $k$ elements. We study the total-variation mixing time of the $S_k$ shuffle when the number of cards $N$ goes to infinity, allowing also $k=k(N)$ to grow with $N$. In particular, we show that the cutoff phenomenon occurs when $k=o(N^{\frac{1}{6}})$.

math.PR↗

arXiv2023

Mixing times for the TASEP in the maximal current phase

We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a segment of size $N$ with open boundaries. We focus on the maximal current phase, and prove that the mixing time is of order $N^{3/2}$, up to logarithmic corrections. In the triple point, where the TASEP with open boundaries approaches the Uniform distribution on the state space, we show that the mixing time is precisely of order $N^{3/2}$. This is conjectured to be the correct order of the mixing time for a wide range of particle systems with maximal current. Our arguments rely on a connection to last-passage percolation, and recent results on moderate deviations of last-passage times.

math.PR↗

arXiv2023

Random planar trees and the Jacobian conjecture

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.

math.CO↗

arXiv2023

Biased random walk on dynamical percolation

We study biased random walks on dynamical percolation on $\mathbb{Z}^d$. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for $d=1$ the speed is increasing, we show that in general this fails in dimension $d \geq 2$. As our main result, we establish two regimes of parameters, separated by an explicit critical curve, such that the speed is either eventually strictly increasing or eventually strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical percolation cluster, where the speed is known to be eventually zero.

math.PR↗

arXiv2022

On the number and size of Markov equivalence classes of random directed acyclic graphs

In causal inference on directed acyclic graphs, the orientation of edges is in general only recovered up to Markov equivalence classes. We study Markov equivalence classes of uniformly random directed acyclic graphs. Using a tower decomposition, we show that the ratio between the number of Markov equivalence classes and directed acyclic graphs approaches a positive constant when the number of sites goes to infinity. For a typical directed acyclic graph, the expected number of elements in its Markov equivalence class remains bounded. More precisely, we prove that for a uniformly chosen directed acyclic graph, the size of its Markov equivalence class has super-polynomial tails.

math.PR↗