arXiv subjects

P. L. Krapivsky

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

At least 37 records · Page 2Linked to original sources

arXiv2025

Dynamic Space Filling

Dynamic space filling (DSF) is a stochastic process defined on any connected graph. Each vertex can host an arbitrary number of particles forming a pile, with every arriving particle landing on the top of the pile. Particles in a pile, except for the particle at the bottom, can hop to neighboring vertices. Eligible particles hop independently and stochastically, with the overall hopping rate set to unity. When the number of vertices in a graph is equal to the total number of particles, the evolution stops when a single particle occupies every vertex. We determine the halting time distribution on complete graphs. Using the mapping of the DSF into a two-species annihilation process, we argue that on $ d$-dimensional tori with $N\gg 1$ vertices, the average halting time scales with the number of vertices as $N^{4/d}$ when $d\leq 4$ and as $N$ when $d>4$.

math.PR↗

arXiv2025

Deterministic simplicial complexes

We investigate simplicial complexes deterministically growing from a single vertex. In the first step, a vertex and an edge connecting it to the primordial vertex are added. The resulting simplicial complex has a 1-dimensional simplex and two 0-dimensional faces (the vertices). The process continues recursively: On the $n$-th step, every existing $d-$dimensional simplex ($d\leq n-1$) joins a new vertex forming a $(d+1)-$dimensional simplex; all $2^{d+1}-2$ new faces are also added so that the resulting object remains a simplicial complex. The emerging simplicial complex has intriguing local and global characteristics. The number of simplices grows faster than $n!$, and the upper-degree distributions follow a power law. Here, the upper degree (or $d$-degree) of a $d$-simplex refers to the number of $(d{+}1)$-simplices that share it as a face. Interestingly, the $d$-degree distributions evolve quite differently for different values of $d$. We compute the Hodge Laplacian spectra of simplicial complexes and show that the spectral and Hausdorff dimensions are infinite. We also explore a constrained version where the dimension of the added simplices is fixed to a finite value $m$. In the constrained model, the number of simplices grows exponentially. In particular, for $m=1$, the spectral dimension is $2$. For $m=2$, the spectral dimension is finite, and the degree distribution follows a power law, while the $1$-degree distribution decays exponentially.

math.CO↗

arXiv2025

Expansion into the vacuum of stochastic gases with long-range interactions

We study the evolution of a system of many point particles initially concentrated in a small region in $d$ dimensions. Particles undergo overdamped motion caused by pairwise interactions through the long-ranged repulsive $r^{-s}$ potential; each particle is also subject to Brownian noise. When $s<d$, the expansion is governed by non-local hydrodynamic equations. In the one-dimensional case, we deduce self-similar solutions for all $s\in (-2,1)$. The expansion of Coulomb gases remains well-defined in the infinite-particle limit: The density is spatially uniform and inversely proportional to time independent of the spatial dimension.

cond-mat.stat-mech↗

arXiv2025

Finite-time blowup of a Brownian particle in a repulsive potential

We consider a Brownian particle performing an overdamped motion in a power-law repulsive potential. If the potential grows with the distance faster than quadratically, the particle escapes to infinity in a finite time. We determine the average blowup time and study the probability distribution of the blowup time. In particular, we show that the long-time tail of this probability distribution decays purely exponentially, while the short-time tail exhibits an essential singularity. These qualitative features turn out to be quite universal, as they occur for all rapidly growing power-law potentials in arbitrary spatial dimensions. The quartic potential is especially tractable, and we analyze it in more detail.

cond-mat.stat-mech↗

arXiv2024

Universal Dynamics of a Passive Particle Driven by Brownian Motion

We investigate the overdamped dynamics of a `passive' particle driven by nonreciprocal interaction with a `driver' Brownian particle. When the interaction between them is short-ranged, the long-time behavior of the driven particle is remarkably universal -- the mean-squared displacement (MSD) and the typical position of the driven particle exhibits the same qualitative behaviors independent of the specific form of the potential. In particular, the MSD grows as $t^{1/2}$ in one dimension and $\log t$ in two spatial dimensions. We compute the exact scaling functions for the position distribution in $d=1$ and $d=2$. These functions are universal when the interaction is short-ranged. For long-ranged interactions, the MSD of the driven particle grows as $t^ϕ$ with exponent $ϕ$ depending on the tail of the potential.

cond-mat.stat-mech↗

arXiv2024

Hiring Strategies

We investigate the hiring problem where a sequence of applicants is sequentially interviewed, and a decision on whether to hire an applicant is immediately made based on the applicant's score. For the maximal and average improvement strategies, the decision depends on the applicant's score and the scores of all employees, i.e., previous successful applicants. For local improvement strategies, an interviewing committee randomly chosen for each applicant makes the decision depending on the score of the applicant and the scores of the members of the committee. These idealized hiring strategies capture the challenges of decision-making under uncertainty. We probe the average score of the best employee, the probability of hiring all first $N$ applicants, the fraction of superior companies in which, throughout the evolution, every hired applicant has a score above expected, etc.

cs.GT↗

arXiv2024

Random sequential covering of a one-dimensional lattice by $k$-mers

In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts that increase coverage are accepted. The process continues indefinitely on an infinite substrate, and we analyze the dynamics of random sequential covering of $\mathbb{Z}$ using $k$-mers. We introduce a method that provides a comprehensive solution to the dynamics of this process. We derive explicit solutions for trimers, tetramers, and pentamers; we study numerically random sequential covering by longer polymers ($k>5$).

cond-mat.stat-mech↗

arXiv2024

Current fluctuations in the Dyson Gas

We study large fluctuations of the current in a Dyson gas, a 1D system of particles interacting through a logarithmic potential and subjected to random noise. We adapt the macroscopic fluctuation theory to the Dyson gas and derive two coupled partial differential equations describing the evolution of the density and momentum. These equations are nonlinear and non-local, and the `boundary' conditions are mixed: some at the initial time and others at the final time. If the initial condition can fluctuate (annealed setting), this boundary-value problem is tractable. We compute the cumulant generating function encoding all the cumulants of the current.

cond-mat.stat-mech↗

arXiv2024

Templating Aggregation

We introduce an aggregation process based on \emph{templating}, where a specified number of constituent clusters must assemble on a larger aggregate, which serves as a scaffold, for a reaction to occur. A simple example is a dimer scaffold, upon which two monomers meet and create another dimer, while dimers and larger aggregates undergo in irreversible aggregation with mass-independent rates. In the mean-field approximation, templating aggregation has unusual kinetics in which the cluster and monomer densities, $c(t)$ and $m(t)$ respectively, decay with time as $c\sim m^2\sim t^{-2/3}$. These starkly contrast to the corresponding behaviors in conventional aggregation, $c\sim \sqrt{m}\sim t^{-1}$. We then treat three natural extensions of templating: (a) the reaction in which $L$ monomers meet and react on an $L$-mer scaffold to create two $L$-mers, (b) multistage scaffold reactions, and (c) templated ligation, in which clusters of all masses serve as scaffolds and binary aggregation is absent.

cond-mat.soft↗

arXiv2024

Error-induced extinction in a multi-type critical birth-death process

Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth-death processes with $n$ distinct types as a birth-death model of EEX in a growing population. Each type-$i$ cell divides independently $(i)\to(i)+(i)$ or mutates $(i)\to(i+1)$ at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-$n$ appears, which cell type can only die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general $n$-type critical birth-death process. We find that the mass function of the number of cells of type-$k$ has algebraic and stationary tail $(\text{size})^{-1-χ_k}$, with $χ_k=2^{1-k}$, for $k=2,\dots,n$, in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability $(\text{time})^{-χ_n}$. We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.

q-bio.PE↗

arXiv2024

Catalytic Coagulation

We introduce an autocatalytic aggregation model in which the rate at which two clusters merge to form a cluster is controlled by the presence of a third "catalytic" cluster whose mass must equal to the mass of one of the reaction partners. The catalyst is unaffected by the joining event and is available to either participate in or catalyze subsequent reactions. This model is meant to mimic the self-replicating reactions that occur in models for the origin of life. We solve the kinetics of this catalytic coagulation model for the case of mass-independent rates and show that the total cluster density decays as $t^{-1/3}$, while the density of clusters of any fixed mass decays as $t^{-2/3}$. These behaviors contrast with the corresponding $t^{-1}$ and $t^{-2}$ scalings for classic aggregation. We extend our model to mass-dependent reaction rates, to situations where only "magic" mass clusters can catalyze reactions, and to include steady monomer input.

cond-mat.stat-mech↗

arXiv2024

Charged Aggregation

We introduce an aggregation process that begins with equal concentrations of positively and negatively `charged' monomers. Oppositely charged monomers merge to form neutral dimers. These dimers are the seeds for subsequent aggregation events in which neutral clusters of necessarily even mass join irreversibly to form neutral aggregates of ever-increasing size. In the mean-field approximation with mass independent reaction rates, we solve for the reaction kinetics and show that the concentration of clusters of mass $k$, $c_k(t)$, asymptotically scales as $A_k/t$, with $A_k$ having a non-trivial dependence on $k$. We also investigate the phenomenon of gelation in charged aggregation when the reaction rate equals the product of the two incident cluster masses. Finally, we generalize our model to the case of three and more types of monomers.

cond-mat.soft↗

arXiv2024

Gelation in input-driven aggregation

We investigate irreversible aggregation processes driven by a source of small mass clusters. In the spatially homogeneous situation, a well-mixed system is consists of clusters of various masses whose concentrations evolve according to an infinite system of nonlinear ordinary differential equations. We focus on the cluster mass distribution in the long time limit. An input-driven aggregation with rates proportional to the product of merging partners undergoes a percolation transition. We examine this process analytically and numerically. There are two theoretical schemes and two natural ways of numerical integration on the level of a truncated system with a finite number of equations. After the percolation transition, the behavior depends on the adopted approach: The giant component quickly engulfs the entire system (Flory approach), or a non-trivial stationary mass distribution emerges (Stockmayer approach). We also outline generalization to ternary aggregation.

cond-mat.stat-mech↗

arXiv2024

First-Passage Duality

We show that the distribution of times for a diffusing particle to first hit an absorber is \emph{independent} of the direction of an external flow field, when we condition on the event that the particle reaches the target for flow away from the target. Thus, in one dimension, the average time for a particle to travel to an absorber a distance $\ell$ away is $\ell/|v|$, independent of the sign of $v$. This duality extends to all moments of the hitting time. In two dimensions, the distribution of first-passage times to an absorbing circle in the radial velocity field $v(r)=Q/(2πr)$ again exhibits duality. Our approach also gives a new perspective on how varying the radial velocity is equivalent to changing the spatial dimension, as well as the transition between transience and strong transience in diffusion.

cond-mat.stat-mech↗

arXiv2024

Impurity in a zero-temperature three-dimensional Fermi gas

We consider an impurity in a sea of zero-temperature fermions uniformly distributed throughout the space. The impurity scatters on fermions. On average, the momentum of impurity decreases with time as $t^{-1/(d+1)}$ in $d$ dimensions, and the momentum distribution acquires a scaling form in the long time limit. We solve the Lorentz-Boltzmann equation for the scaled momentum distribution of the impurity in three dimensions. The solution is a combination of confluent hypergeometric functions. In two spatial dimensions, the Lorentz-Boltzmann equation is analytically intractable, so we merely extract a few exact predictions about asymptotic behaviors when the scaled momentum of the impurity is small or large.

cond-mat.quant-gas↗

arXiv2024

Random Maps with Sociological Flavor

A map of a set to itself admits a representation by a graph with vertices being the elements of the set and an edge between every vertex and its image. Communities defined as the maximal connected components are uni-cyclic. The distributions of the sizes of communities and lengths of cycles for unconstrained random maps is a classical subject. We call experts the images and followers the remaining vertexes, and we further define prophets, egocentrics, and introverts. We introduce and analyze classes of random maps with sociological flavor.

math.CO↗

arXiv2024

Simple evolving random graphs

We study the evolution of graphs densifying by adding edges: Two vertices are chosen randomly, and an edge is (i) established if each vertex belongs to a tree; (ii) established with probability $p$ if only one vertex belongs to a tree; (iii) an attempt fails if both vertices belong to unicyclic components. Emerging simple random graphs contain only trees and unicycles. In the thermodynamic limit of an infinite number of vertices, the fraction of vertices in unicycles undergoes a phase transition resembling a percolation transition in classical random graphs. In classical random graphs, a complex giant component born at the transition eventually engulfs all finite components and densifies forever. The evolution of simple random graphs freezes when trees disappear. We quantify simple random graphs in the supercritical phase and the properties of the frozen state.

math.PR↗

arXiv2023

Supercluster states and phase transitions in aggregation-fragmentation processes

We study the evolution of aggregates triggered by collisions with monomers that either lead to the attachment of monomers or the break-up of aggregates into constituting monomers. Depending on parameters quantifying addition and break-up rates, the system falls into a jammed or a steady state. Supercluster states (SCSs) are very peculiar non-extensive jammed states that also arise in some models. Fluctuations underlie the formation of the SCSs. Conventional tools, such as the van Kampen expansion, apply to small fluctuations. We go beyond the van Kampen expansion and determine a set of critical exponents quantifying SCSs. We observe continuous and discontinuous phase transitions between the states. Our theoretical predictions are in good agreement with numerical results.

cond-mat.stat-mech↗