arXiv subjects

P. L. Krapivsky

Publications and source records attributed to P. L. Krapivsky.

At least 19 recordsLinked to original sources

arXiv2026

The Riviera model with egoistical settlers

The Riviera model mimics a densifying settlement along the coastline. In the lattice version, houses are built sequentially in empty sites with the constraint that every newly built house has at least one empty neighboring site. The distribution of clusters of adjacent houses does not obey a closed set of evolutionary equations, but the void-cluster-void distribution does. We compute the latter and extract the cluster distribution from it. In the jammed state, when all voids have length one, the cluster distribution takes a simple closed form and exhibits factorial decay with cluster size. For finite systems, we use a static approach that directly analyzes jammed states. If the coastline is a finite segment, we determine the statistics of the number of empty sites in the jammed state (the average, variance, and higher cumulants). We also study a continuum version in which houses are built along the line, subject to a minimum separation from at least one neighboring house.

cond-mat.stat-mech↗

arXiv2026

The Winner Takes It All

The winner-takes-all process takes place on an arbitrary graph. There is an agent on each vertex of the graph, and active agents at neighboring vertices play games. In each game, a randomly chosen agent wins, while the loser is eliminated from subsequent games. The games are played at random times, finishing instantaneously, and ceasing when each active agent has only losers among its neighbors. On the one-dimensional lattice, the fraction of winners in the final state is $e^{-1}$; we also determine the fractions $w_j$ of winners who won $j=0, 1, 2$ games. For finite segments, we determine statistics of the total number of winners (the average, the variance, and all higher cumulants), the probabilities of attaining the minimum or maximum possible number of winners, and establish the behavior near the boundaries. For infinite regular trees with vertices of degree $d$, i.e., Bethe lattices with coordination number $d$, we show that the fraction of winners is $(2/d)^{d/(d-2)}$.

physics.soc-ph↗

arXiv2026

Random Recursive Simplicial Complexes

We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an existing simplex is uniform among simplices of dimension $<m$, the number $S_d$ of simplices of any admissible dimension $d\leq m$ is an asymptotically self-averaging random variable. This feature allows us to determine the asymptotic growth law of the average of $S_d$ when the number of vertices diverges. We also probe the degree distribution, examine the probabilities of various extreme outcomes, and analyze the characteristics of the first vertex.

math.CO↗

arXiv2026

The rank and layer distributions in random recursive trees

The distribution of node depths in a network is crucial for analyzing network structure. Two measures, rank and layer, quantify how deep inside a network a node is. The rank is the node's distance to the closest leaf, a node of degree 1. The layer is the number of times all leaves must be stripped off for the node to become a leaf. We derive exact recursive expressions for the rank and layer distributions in random recursive trees. We show that the rank and layer distributions decay factorially and geometrically, respectively, and prove the self-averaging of the numbers of nodes with fixed rank or layer. Unlike previous studies, our approach does not depend on labels or a root node, providing a more versatile framework for analyzing rank and layer distributions in complex networks.

math.PR↗

arXiv2026

Universal fluctuations of first discoveries in competitive exploration

Random exploration is usually quantified by how fast new space is found, from the range of a single walker to the territory collectively covered by many walkers. In competitive exploration, first arrival secures an exclusive resource, as when foragers compete for food items or agents capture distributed targets. It is then no longer enough to know which sites have been discovered: one must determine, for each discovered site, which searcher reached it first. We introduce the discovery share $X_n$, the fraction of the first $n$ collective discoveries secured by a tagged searcher. For two identical competitors, exchange symmetry fixes $\langle X_n\rangle=1/2$, but the central question is whether this equal split emerges in each long exploration history or only on average, i.e. whether early competitive advantages are erased or persist. Here we show that the answer is controlled by the spectral dimension $d_s$, defined by the large-time decay of the probability that a single searcher is at its starting point after $t$ steps, $p_0(t)\sim t^{-d_s/2}$. Across ordinary diffusion, long-range superdiffusion and subdiffusion induced by crowding or memory, $d_s$ separates persistent randomness in recurrent exploration $(d_s<2)$, anomalously slow non-Gaussian concentration for $2\le d_s<3$, and Gaussian concentration, logarithmically corrected at $d_s=3$, for $d_s\ge3$. For $d_s\ge2$, we derive exact asymptotic variances, including prefactors, and the discovery scale on which competitive imbalances are erased. Two-point correlations of first-discovery labels identify the memory mechanism behind these regimes. The same phase structure persists under changes in geometry, competitor heterogeneity, number of competitors and memory, revealing a general fluctuation theory of first-arrival inequalities.

cond-mat.stat-mech↗

arXiv2026

Mixed phases in feedback Ising models

We study mean-field Ising models in which the coupling depends on the magnetization via a feedback function. We identify mixed phases (MPs) and show that they can be stable at zero temperature for sufficiently strong feedback. Moreover, stable MPs are always super-stable, meaning that perturbations decay linearly in time. Feedback Ising models (FIMs) provide a useful framework for phase transformations between aligned phases via stable and unstable intermediate phases in multistable systems. We also analyze the dynamical behavior of FIMs driven by a varying magnetic field and discuss basic properties of finite-dimensional FIMs.

cond-mat.stat-mech↗

arXiv2026

Regularized products of Gauss and Eisenstein integers and primes

We provide heuristic computations à la Euler of the regularized infinite products of Gauss and Eisenstein integers and primes. Our approach, yielding explicit expressions, is inspired by the work by Muñoz García and Pérez-Marco, who evaluated the product of all natural primes to $4π^2$.

math.NT↗

arXiv2026

Zeta-regularization and natural boundaries: Sums and products of integers and primes

Euler regularized the divergent product of all natural numbers and found beautiful formulas for regularized sums of integer powers of natural numbers. These derivations essentially relied on what is now called the zeta-regularization technique, although analytical continuation had not yet been invented. This classic method is however not applicable to the product of all primes, as the prime zeta function has a natural boundary along the imaginary axis. Muñoz García and Pérez-Marco overcame this obstacle and evaluated the product of all primes to $4π^2$ by finding an appropriately regularized value of the derivative of the prime zeta function at the origin, lying on the natural boundary. We extend their approach in two novel directions. First, we show how to make sense of the sum of all primes. This regularization requires going a finite distance beyond the natural boundary. Second, we determine the regularized products of integers and primes in the nine imaginary quadratic fields where integers have a unique factorization into primes, and establish a general power-law relationship between products of integers and primes. Two well-known examples are Gauss and Eisenstein integers. The interest in this approach goes beyond number theory. In a variety of physical situations, the zeta-regularization technique is indeed not applicable because the relevant zeta function has a natural boundary.

math.NT↗

arXiv2026

Aggregation-Fragmentation Processes with Broken Detailed Balance

We study aggregation-fragmentation processes in which pairs of clusters can aggregate, and each cluster can break into two fragments. If the rates of aggregation and fragmentation do not depend on the masses, detailed balance does not hold, but nonequilibrium steady states can still be deduced from an exact solution for the Laplace transform. For models in which aggregation rates remain constant but fragmentation rates scale as $(\text{mass})^β$, detailed balance holds only when $β=1$. Away from this solvable case, we employ asymptotic techniques and show that when $β\geq 0$, the steady states share similarities with those from the mass-independent ($β=0$) model. An instantaneous shattering transition with continuous mass loss occurs when $β<0$.

cond-mat.stat-mech↗

arXiv2026

Anomalous scaling in redirection networks

In networks that grow by isotropic redirection (IR), a new node selects an initial target node uniformly at random and attaches to a randomly chosen neighbor of the target. The emerging networks exhibit leaf proliferation, in which the number of nonleaves scales sublinearly as $N^μ$ and the degree distribution has an algebraic tail with exponent $1+μ$. To understand these mysterious properties, we introduce a class of models with redirection to leaves whenever possible. The resulting networks exhibit qualitatively similar phenomenology to IR networks, but avoid the inherent non-locality of the IR growth rule. These networks admit an analytical description of the leaf degree distribution, from which we extract the exponent $μ$.

cond-mat.stat-mech↗

arXiv2026

Dynamics of feedback Ising model

We study the dynamics of a mean-field Ising model whose coupling depends on the magnetization via a linear feedback function. A key feature of this linear feedback Ising model (FIM) is the possibility of temperature-induced bistability, where a temperature increase can favor bistability between two phases. We show that the linear FIM provides a minimal model for a transcritical bifurcation as the temperature varies. Moreover, there can be two or three critical temperatures when the external magnetic field is non-negative. In the bistable region, we identify a Maxwell temperature where the two phases are equally probable, and we find that increasing the temperature favors the lower phase. We show that the probability distribution becomes non-Gaussian on certain time intervals when the magnetization converges algebraically at either zero temperature or critical temperatures. Near critical points in the parameter space, we derive a Fokker-Planck equation, construct the families of equilibrium distributions, and formulate scaling laws for transition rates between two stable equilibria. The linear FIM offers considerable flexibility in controlling steady-state bifurcations and their associated equilibrium distributions, which can be desirable for modeling feedback systems across various disciplines.

cond-mat.stat-mech↗

arXiv2026

Self-reinforcing cascades: A spreading model for beliefs or products of varying intensity or quality

Models of how things spread often assume that transmission mechanisms are fixed over time. However, social contagions--the spread of ideas, beliefs, innovations--can lose or gain in momentum as they spread: ideas can get reinforced, beliefs strengthened, products refined. We study the impacts of such self-reinforcement mechanisms in cascade dynamics. We use different mathematical modeling techniques to capture the recursive, yet changing nature of the process. We find a critical regime with a range of power-law cascade size distributions with non-universal scaling exponents. This regime clashes with classic models, where criticality requires fine tuning at a precise critical point. Self-reinforced cascades produce critical-like behavior over a wide range of parameters, which may help explain the ubiquity of power-law distributions in empirical social data.

physics.soc-ph↗

arXiv2026

Leaves of preferential attachment trees

We provide a local probabilistic description of the limiting statistics of large preferential attachment trees in terms of the ordinary degree (number of neighbors) but augmented with information on leafdegree (number of neighbors that are leaves). The full description is the joint degree-leafdegree distribution $n_{k,\ell}$, which we derive from its associated multivariate generating function. From $n_{k,\ell}$ we obtain the leafdegree distribution, $m_{\ell}$, as well as the fraction of vertices that are protected (nonleaves with leafdegree zero) as a function of degree, $n_{k,0}$, among numerous other results. We also examine fluctuations and concentration of joint degree-leafdegree empirical counts $N_{k,\ell}$. Although our main findings pertain to the preferential attachment tree, the approach we present is highly generalizable and can characterize numerous existing models, in addition to facilitating the development of tractable new models. We further demonstrate the approach by analyzing $n_{k,\ell}$ in two other models: the random recursive tree, and a redirection-based model.

cond-mat.stat-mech↗

arXiv2026

Impurity dynamics in a zero-temperature gas

If energy is suddenly released in a localized region of space uniformly filled with identical stationary hard spheres, the outcome is a blast with an asymptotically spherical shock wave separating moving and stationary hard spheres. The radius $R(t)$ of the region filled with the moving spheres grows as $t^{2/(d+2)}$, where $d$ is the spatial dimension. The simplest way to inject energy is to kick a few `impurity' particles. Using hydrodynamics and kinetic theory, we argue that the typical displacement of an impurity scales as $R_{\rm imp} \sim λ(R/λ)^{(4+3d^2)/(8+3d^2)}$, where $λ$ is the mean-free path in the initial state. The number of collisions experienced by each impurity grows as $(R/λ)^{(8+2d^2)/(8+3d^2)}$, while its average speed decreases as $t^{-d(8-2d+3d^2)/[(2+d)(8+3d^2)]}$. In $2D$, the predictions for impurity displacement, collision numbers, and speed are $t^{2/5},~t^{2/5}$ and $t^{-2/5}$, respectively. These predictions are in reasonable agreement with the results of molecular dynamics simulations.

cond-mat.stat-mech↗

arXiv2025

Finite-time consensus in a compromise process

A compromise process describes the evolution of opinions through binary interactions. Opinions are real numbers, and at each step, two randomly selected agents reach a compromise by averaging their pre-interaction opinions. We prove that if the number $N$ of agents is a power of two, then consensus emerges after a finite number of compromise events with probability one; otherwise, consensus cannot be reached in a finite number of steps, provided the initial opinions are in a general position. The number of steps required to reach consensus is random for $N=2^k$ with $k\geq 2$. We prove that the smallest number of steps is $k\cdot 2^{k-1}$ when the initial opinions are in a general position. For $N=4$, we determine the distribution of the number of steps. In particular, we show that it has a purely exponential tail and compute all cumulants.

physics.soc-ph↗

arXiv2025

Statistics of leaves in growing random trees

Leaves, i.e., vertices of degree one, can play a significant role in graph structure, especially in sparsely connected settings in which leaves often constitute the largest fraction of vertices. We consider a leaf-based counterpart of the degree, namely, the leaf degree -- the number of leaves a vertex is connected to -- and the associated leaf degree distribution, analogous to the degree distribution. We determine the leaf degree distribution of random recursive trees (RRTs) and trees grown via a leaf-based preferential attachment mechanism that we introduce. The RRT leaf degree distribution decays factorially, in contrast with its purely geometric degree distribution. In the one-parameter leaf-based growth model, each new vertex attaches to an existing vertex with rate $\ell$ + a, where $\ell$ is the leaf degree of the existing vertex, and a > 0. The leaf degree distribution has a powerlaw tail when 0 < a < 1 and an exponential tail (with algebraic prefactor) for a > 1. The critical case of a = 1 has a leaf degree distribution with stretched exponential tail. We compute a variety of additional characteristics in these models and conjecture asymptotic equivalence of degree and leaf degree powerlaw tail exponent in the scale free regime. We highlight several avenues of possible extension for future studies.

cond-mat.stat-mech↗

arXiv2025

Growing unlabeled networks

Models of growing networks are a central topic in network science. In these models, vertices are usually labeled by their arrival time, distinguishing even those node pairs whose structural roles are identical. In contrast, unlabeled networks encode only structure, so unlabeled growth rules must be defined in terms of structurally distinguishable outcomes; network symmetries therefore play a key role in unlabeled growth dynamics. Here, we introduce and study models of growing unlabeled trees, defined in analogy to widely-studied labeled growth models such as uniform and preferential attachment. We develop a theoretical formalism to analyze these trees via tracking their leaf-based statistics. We find that while many characteristics of labeled network growth are retained, numerous critical differences arise, caused primarily by symmetries among leaves in common neighborhoods. In particular, degree heterogeneity is enhanced, with the strength of this enhancement depending on details of growth dynamics: mild enhancement for uniform attachment, and extreme enhancement for preferential attachment. These results and the developed analytical formalism may be of interest beyond the setting of growing unlabeled trees.

physics.soc-ph↗

arXiv2025

A generative model of function growth explains hidden self-similarities across biological and social systems

From genomes and ecosystems to bureaucracies and cities, the growth of complex systems occurs by adding new types of functions and expanding existing ones. We present a simple generative model that generalizes the Yule-Simon process by including: (i) a size-dependent probability of introducing new functions, and (ii) a generalized preferential attachment mechanism for expanding existing ones. We uncover a shared underlying structure that helps explain how function diversity evolves in empirical observations, such as prokaryotic proteomes, U.S. federal agencies, and urban economies. We show that real systems are often best represented as having non-Zipfian rank-frequency distributions, driven by sublinear preferential attachment, whilst still maintaining power-law scaling in their abundance distributions. Furthermore, our analytics explain five distinct phases of the organization of functional elements across complex systems. The model integrates empirical findings regarding the logarithmic growth of diversity in cities and the self-similarity of their rank-frequency distributions. Self-similarity previously observed in the rank-frequency distributions of cities is not observed in cells and federal agencies -- however, under a rescaling relative to the total diversity, all systems admit self-similar structures predicted by our theory.

physics.soc-ph↗