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Stjepan Šebek

Publications and source records attributed to Stjepan Šebek.

At least 19 recordsLinked to original sources
arXiv2026

Generalised Random Parking for Trapeziums on a Strip

In this article, we study a generalisation of Rényi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.

math.PR↗
arXiv2026

Improved bounds on the inradius and inverse inradius process of the convex hull of planar Brownian motion

We study the inradius and inverse inradius of the convex hull of standard planar Brownian motion. Our main goals are to improve the existing bounds on the expected values of these quantities and to establish variance bounds. The main contributions of the paper, however, are the following intermediate results we obtain in order to compute our bounds. We find closed-form expressions for the expected values of the minimum and maximum of the ranges of two independent one-dimensional Brownian motions. Besides improving the upper bound on the expected inradius, this result also gives an explicit expression for a quantity that has been investigated repeatedly in the literature and for which only a numerical evaluation has been known so far. Furthermore, we study the exit time of planar Brownian motion from an unbounded region that we call the hourglass domain and compute its first two moments. This result leads to a nearly fivefold improvement of the previous upper bound on the expected inverse inradius and plays an essential role in establishing the corresponding upper bound on the variance.

math.PR↗
arXiv2026

Fluctuations for diameter and perimeter of convex hulls of multiple random walks

We study the diameter and perimeter of the convex hull generated by finitely many independent planar random walks whose increments have finite second moments. The large-time fluctuations are governed by the geometry of the polygon formed by the drift vectors. We develop an $L^2$-approximation framework, based on Wald-type maximal central limit theorems, which reduces the asymptotic analysis of the hull to a finite collection of endpoint, maximal-projection, and Brownian support-function terms. For the diameter, we obtain general max-type limit theorems, Gaussian in the case of a unique extremal diametrical pair and typically non-Gaussian when several extremal pairs compete. For the perimeter, we prove a general distributional limit: non-zero extremal drifts contribute maxima of Gaussian projections, while zero-drift extremal walks contribute Brownian support-function terms. The results recover the previously known Gaussian regimes (the case of one or two walks) and identify the non-Gaussian limits in the degenerate and boundary cases left open (even for two walks). We also give $L^2$ approximations of the convex hull by simpler random sets, under Hausdorff and $\ell_1$ metrics on compact convex sets. Our proofs work under the optimal finite second moment assumption.

math.PR↗
arXiv2026

Weighted-threshold Coupon Collection

We study a weighted-threshold version of the coupon collector problem in continuous time. Each type $i$ is discovered at rate $λp_i$ and, once discovered, contributes weight $w_i$, where $p$ and $w$ are probability vectors. The stopping time when the total weight of the discovered types first exceeds a fixed threshold $θ\in (0,1)$ is called the quorum time. We first prove concentration estimates and compare the quorum time with the corresponding deterministic threshold time obtained from the mean discovered weight. When all discovery rates are equal and the largest individual weight tends to zero, the first-order asymptotics are universal and do not depend on the weight vector. We then analyze the aligned Zipf family $p_i = w_i \propto i^{-s}$. This model has three regimes: a deterministic linear scale for $0\le s < 1$, a critical scale $H_NN^θ$ at $s=1$, with an explicit leading constant, and a non-degenerate random hitting-time limit for $s>1$. Finally, we show that the expected quorum time need not be monotone in the Zipf exponent.

math.PR↗
arXiv2026

Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk

We study the convex hull of planar Brownian motion run until the exit time from the unit disk. Our primary objectives are to compute the expected perimeter and expected area of this convex hull, thereby complementing recent results on the convex hull of reflecting Brownian motion in confined geometries. We reduce the problem of computing the expected perimeter to computing the expected value of the Brownian motion's maximum horizontal displacement at the exit time, and then recast this maximum in terms of harmonic measure in a domain we call the truncated disk. The problem of computing the expected area is reduced to computing the expected value of the difference of squares of the Brownian motion's maximum horizontal displacement at the exit time, and the value of the vertical displacement at the time this maximum horizontal displacement is achieved. In particular, we obtain exact expressions for both the expected perimeter and the expected area. We conclude with further results on the expected areas of two related hulls of the Brownian path run until exiting the disk, namely, the star hull and topological hull.

math.PR↗
arXiv2026

On the convex hull of a planar Brownian bridge with a random Gaussian endpoint

We consider a one-parameter family of isotropic planar Gaussian processes \[ X_σ(t) =B_t+σt Z,\qquad 0\le t\le 1,\quad 0\le σ\le 1, \] where $B$ is a standard ($0$-to-$0$) planar Brownian bridge on $[0,1]$, and $Z\sim \mathrm N(0,I)$ is a standard Gaussian random vector independent of $B$. The family interpolates between standard planar Brownian bridge ($σ=0$) and standard planar Brownian motion ($σ=1$). As the main result of the paper we compute the expected perimeter and area of the convex hull of the random set $\left\{X_σ(t) \colon 0\le t\le 1\right\}$ as closed formulas in terms of $σ$, and recover the classical Brownian bridge and Brownian motion values at $σ=0$ and $σ=1$. We also consider the convex hull spanned by multiple independent processes of this type and the possibilities for closed formulas in special cases. The key observation in our argument is that the isotropy property reduces the expected perimeter and area to one-dimensional quantities through the support function and Cauchy's formulas.

math.PR↗
arXiv2025

On the convex hull of two planar random walks

In this paper, we study the limiting behavior of the perimeter and diameter functionals of the convex hull spanned by the first $n$ steps of two planar random walks. As the main results, we obtain the strong law of large numbers and the central limit theorem for the perimeter and diameter of these random sets.

math.PR↗
arXiv2024

A model of random sequential adsorption on a ladder graph

In random sequential adsorption (RSA), objects are deposited on a substrate randomly, irreversibly, and sequentially. Attempts of deposition that lead to an overlap with previously deposited objects are discarded. The process continues until the system reaches a jammed state when no further additions are possible. We analyze a class of RSA models on a two-row square ladder graph in which landing on an empty site in a graph is allowed when at least $b$ neighboring sites in the graph are unoccupied ($b \in \mathbb{N}$). In this paper we complement this typical way of studying RSA models by analyzing also the structure of the set of all jammed states in a static way, disregarding the dynamics that led to a particular jammed state. In both considered settings (dynamic and static) we provide explicit expressions for key statistics that describe the average proportion of the substrate covered by deposited objects, and then we comment on significant differences between the two settings. We illustrate all of our findings through a toy model for ensembles of trapped Rydberg atoms with blockade range $b$.

cond-mat.stat-mech↗
arXiv2024

Bounds on the size of the convex hull of planar Brownian motion and related inverse processes

We establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected values of the perimeter and the area of the Brownian convex hull are known explicitly, and satisfactory bounds on the expected value of its diameter can be found in the literature as well. In this work we investigate circumradius and inradius of the Brownian convex hull and obtain lower and upper bounds on their expected values. Our other goal is to find bounds on the related inverse processes (that correspond to the perimeter, area, diameter, circumradius and inradius of the convex hull) which provide us with some information on the speed of growth of the size of the Brownian convex hull.

math.PR↗
arXiv2024

Iterated-logarithm laws for convex hulls of random walks with drift

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.

math.PR↗
arXiv2024

Convex hull of Brownian motion and Brownian bridge

In this article we study the convex hull spanned by the union of trajectories of a standard planar Brownian motion, and an independent standard planar Brownian bridge. We find exact values of the expectation of perimeter and area of such a convex hull. As an auxiliary result, that is of interest in its own right, we provide an explicit shape of the probability density function of a random variable that represents the time when combined maximum of a standard one-dimensional Brownian motion, and an independent standard one-dimensional Brownian bridge is attained. At the end, we generalize our results to the case of multiple independent standard planar Brownian motions and Brownian bridges.

math.PR↗
arXiv2024

Predators and altruists arriving on jammed Riviera

The Riviera model is a combinatorial model for a settlement along a coastline, introduced recently by the authors. Of most interest are the so-called jammed states, where no more houses can be built without violating the condition that every house needs to have free space to at least one of its sides. In this paper, we introduce new agents (predators and altruists) that want to build houses once the settlement is already in the jammed state. Their behavior is governed by a different set of rules, and this allows them to build new houses even though the settlement is jammed. Our main focus is to detect jammed configurations that are resistant to predators, to altruists, and to both predators and altruists. We provide bivariate generating functions, and complexity functions (configurational entropies) for such jammed configurations. We also discuss this problem in the two-dimensional setting of a combinatorial settlement planning model that was also recently introduced by the authors, and of which the Riviera model is just a special case.

math.CO↗
arXiv2023

Complexity Function of Jammed Configurations of Rydberg Atoms

In this article, we determine the complexity function (configurational entropy) of jammed configurations of Rydberg atoms on a one-dimensional lattice. Our method consists of providing asymptotics for the number of jammed configurations determined by direct combinatorial reasoning. In this way we reduce the computation of complexity to solving a constrained optimization problem for the Shannon's entropy function. We show that the complexity can be expressed explicitly in terms of the root of a certain polynomial of degree $b$, where $b$ is the so-called blockade range of a Rydberg atom. Our results are put in a relation with the model of irreversible deposition of $k$-mers on a one-dimensional lattice.

math.CO↗
arXiv2023

Learning from non-irreducible Markov chains

Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong correlation between the marginals of the data-generating process, suggesting that the i.i.d. assumption is not always justified. This problem has been already considered in the context of Markov chains satisfying the Doeblin condition. This condition, among other things, implies that the chain is not singular in its behavior, i.e. it is irreducible. In this article, we focus on the case when the training data set is drawn from a not necessarily irreducible Markov chain. Under the assumption that the chain is uniformly ergodic with respect to the $\mathrm{L}^1$-Wasserstein distance, and certain regularity assumptions on the hypothesis class and the state space of the chain, we first obtain a uniform convergence result for the corresponding sample error, and then we conclude learnability of the approximate sample error minimization algorithm and find its generalization bounds. At the end, a relative uniform convergence result for the sample error is also discussed.

math.ST↗
arXiv2022

On a variant of Flory model

We consider a one-dimensional variant of a recently introduced settlement planning problem in which houses can be built on finite portions of the rectangular integer lattice subject to certain requirements on the amount of insolation they receive. In our model, each house occupies a unit square on a $1 \times n$ strip, with the restriction that at least one of the neighboring squares must be free. We are interested mostly in situations in which no further building is possible, i.e. in maximal configurations of houses in the strip. We reinterpret the problem as a problem of restricted packing of vertices in a path graph and then apply the transfer matrix method in order to compute the bivariate generating functions for the sequences enumerating all maximal configurations of a given length with respect to the number of houses. This allows us to determine the asymptotic behavior of the enumerating sequences and to compute some interesting statistics. Along the way, we establish close connections between our maximal configurations and several other types of combinatorial objects, including restricted permutations and walks on certain small oriented graphs. In all cases we provide combinatorial proofs. We then generalize our results in several directions by considering multi-story houses, by varying the insolation restrictions, and, finally, by considering strips of width 2 and 3. At the end we comment on several possible directions of future research.

math.CO↗
arXiv2022

Convex hulls of stable random walks

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.

math.PR↗
arXiv2021

Capacity of the range of random walks on groups

In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem.

math.PR↗