arXiv subjects

Jacob Calvert

Publications and source records attributed to Jacob Calvert.

14 recordsLinked to original sources

arXiv2026

Boltzmann-Like Occupation of Nonequilibrium Steady States on Dense Networks

A central problem in statistical physics is to extend the Boltzmann distribution to nonequilibrium steady states (NESS). We prove that NESS on large dense networks have Boltzmann-like occupation despite extensive entropy production. We further show that the active-matter heuristic of "low rattling" is asymptotically exact. Intuitively, these NESS spend a greater fraction of their time in states they leave more slowly. This explanation extends to the broader class of "equiaccessible" steady states, which play a role in our analysis akin to that of equilibrium in linear response.

cond-mat.stat-mech↗

arXiv2025

How many more is different?

From the formation of ice in small clusters of water molecules to the mass raids of army ant colonies, the emergent behavior of collectives depends critically on their size. At the same time, common wisdom holds that such behaviors are robust to the loss of individuals. This tension points to the need for a more systematic study of how number influences collective behavior. We initiate this study by focusing on collective behaviors that change abruptly at certain critical numbers of individuals. We show that a subtle modification of standard bifurcation analysis identifies such critical numbers, including those associated with discreteness- and noise-induced transitions. By treating them as instances of the same phenomenon, we show that critical numbers across physical scales and scientific domains commonly arise from competing feedbacks that scale differently with number. We then use this idea to find overlooked critical numbers in past studies of collective behavior and explore the implications for their conclusions. In particular, we highlight how deterministic approximations of stochastic models can fail near critical numbers. We close by distinguishing these qualitative changes from density-dependent phase transitions and by discussing how our approach could generalize to broader classes of collective behaviors.

q-bio.PE↗

arXiv2025

Correlation thresholds in the steady states of particle systems and spin glasses

A growing body of theoretical and empirical evidence shows that the global steady-state distributions of many equilibrium and nonequilibrium systems approximately satisfy an analogue of the Boltzmann distribution, with a local dynamical property of states playing the role of energy. The correlation between the effective potential of the steady-state distribution and the logarithm of the exit rates determines the quality of this approximation. We demonstrate and explain this phenomenon in a simple one-dimensional particle system and in random dynamics of the Sherrington-Kirkpatrick spin glass by providing the first explicit estimates of this correlation. We find that, as parameters of the dynamics vary, each system exhibits a threshold above and below which the correlation dramatically differs. We explain how these thresholds arise from underlying transitions in the relationship between the local and global "parts" of the effective potential.

cond-mat.stat-mech↗

arXiv2025

Local-global correlations of dynamics on disordered energy landscapes

The stationary distribution of a continuous-time Markov chain generally arises from a complicated global balance of probability fluxes. Nevertheless, empirical evidence shows that the effective potential, defined as the negative logarithm of the stationary distribution, is often highly correlated with a simple local property of a state: the logarithm of its exit rate. To better understand why and how typically this correlation is high, we study reversible reaction kinetics on energy landscapes consisting of Gaussian wells and barriers, respectively associated with the vertices and edges of regular graphs. We find that for the correlation to be high it suffices for the heights of the barriers to vary significantly less than the depths of the wells, regardless of the degree of the underlying graph. As an application, we bound below the expected correlation exhibited by dynamics of the random energy model, known as the Bouchaud trap model. We anticipate that the proof, which combines a general lower bound of the expected correlation with the Gaussian concentration inequality, can be extended to several other classes of models.

math.PR↗

arXiv2025

On the principal eigenvectors of random Markov matrices

We analyze the invariant distributions of continuous-time and discrete-time random walks on randomly weighted complete digraphs. These distributions correspond to the principal left eigenvectors of the associated random Markov generators and kernels, viewed as random matrices. While much is known about the spectra of these matrices, relatively little is known about the principal left eigenvectors, which are delicate random objects for which no explicit form is known. We consider a broad class of such matrices obtained by associating random weights to the vertices and edges of the complete digraph. Our main result concerns the total variation distance between the invariant distribution of the continuous-time random walk and the distribution that is inversely proportional to the vertex weights. It states that, if the edge weights are i.i.d. with a finite $p$-th moment for some $p>4$, then this distance a.s. converges to zero as the number of vertices grows large, even when the vertex weights are heavy-tailed. We further answer a question of Bordenave, Caputo, and Chafa\"i by showing that, despite the dependence of the entries in the corresponding Markov kernel, its invariant distribution is asymptotically uniform a.s., so long as the edge weights have a finite second moment.

math.PR↗

arXiv2024

The Geometry of Fixed-Magnetization Spin Systems at Low Temperature

Spin systems are fundamental models of statistical physics that provide insight into collective behavior across scientific domains. Their interest to computer science stems in part from the deep connection between the phase transitions they exhibit and the computational complexity of sampling from the probability distributions they describe. Our focus is on the geometry of spin configurations, motivated by applications to programmable matter and computational biology. Rigorous results in this vein are scarce because the natural setting of these applications is the low-temperature, fixed-magnetization regime. Recent progress in this regime is largely limited to spin systems under which magnetization concentrates, which enables the analysis to be reduced to that of the simpler, variable-magnetization case. More complicated models, like those that arise in applications, do not share this property. We study the geometry of spin configurations on the triangular lattice under the Generalized Potts Model (GPM), which generalizes many fundamental models of statistical physics, including the Ising, Potts, clock, and Blume--Capel models. Moreover, it specializes to models used to program active matter to solve tasks like compression and separation, and it is closely related to the Cellular Potts Model, widely used in computational models of biological processes. Our main result shows that, under the fixed-magnetization GPM at low temperature, spins of different types are typically partitioned into regions of mostly one type, separated by boundaries that have nearly minimal perimeter. The proof uses techniques from Pirogov--Sinai theory to extend a classic Peierls argument for the fixed-magnetization Ising model, and introduces a new approach for comparing the partition functions of fixed- and variable-magnetization models.

math-ph↗

arXiv2024

Single Bridge Formation in Self-Organizing Particle Systems

Local interactions of uncoordinated individuals produce the collective behaviors of many biological systems, inspiring much of the current research in programmable matter. A striking example is the spontaneous assembly of fire ants into "bridges" comprising their own bodies to traverse obstacles and reach sources of food. Experiments and simulations suggest that, remarkably, these ants always form one bridge -- instead of multiple, competing bridges -- despite a lack of central coordination. We argue that the reliable formation of a single bridge does not require sophistication on behalf of the individuals by provably reproducing this behavior in a self-organizing particle system. We show that the formation of a single bridge by the particles is a statistical inevitability of their preferences to move in a particular direction, such as toward a food source, and their preference for more neighbors. Two parameters, $\eta$ and $\beta$, reflect the strengths of these preferences and determine the Gibbs stationary measure of the corresponding particle system's Markov chain dynamics. We show that a single bridge almost certainly forms when $\eta$ and $\beta$ are sufficiently large. Our proof introduces an auxiliary Markov chain, called an "occupancy chain," that captures only the significant, global changes to the system. Through the occupancy chain, we abstract away information about the motion of individual particles, but we gain a more direct means of analyzing their collective behavior. Such abstractions provide a promising new direction for understanding many other systems of programmable matter.

cs.DC↗

arXiv2023

A local-global principle for nonequilibrium steady states

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of a configuration. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent work demonstrates that a local property of configurations called "rattling" predicts the global steady states of a broad class of nonequilibrium systems. We interpret this emerging physical theory in terms of Markov processes to generalize and prove its main claims. Surprisingly, we find that the idea at the core of rattling theory is so general as to apply to equilibrium and nonequilibrium systems alike. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global "parts" of an arbitrary steady state. We show how these key quantities characterize the local-global relationships of random walks on random graphs, various spin-glass dynamics, and models of animal collective behavior.

math.PR↗

arXiv2023

Critical numerosity in collective behavior

Natural collectives, despite comprising individuals who may not know their numerosity, can exhibit behaviors that depend sensitively on it. This paper proves that the collective behavior of number-oblivious individuals can even have a critical numerosity, above and below which it qualitatively differs. We formalize the concept of critical numerosity in terms of a family of zero--one laws and introduce a model of collective motion, called chain activation and transport (CAT), that has one. CAT describes the collective motion of $n \geq 2$ individuals as a Markov chain that rearranges $n$-element subsets of the $d$-dimensional grid, $m < n$ elements at a time. According to the individuals' dynamics, with each step, CAT removes $m$ elements from the set and then progressively adds $m$ elements to the boundary of what remains, in a way that favors the consecutive addition and removal of nearby elements. This paper proves that, if $d \geq 3$, then CAT has a critical numerosity of $n_c = 2m+2$ with respect to the behavior of its diameter. Specifically, if $n < n_c$, then the elements form one "cluster," the diameter of which has an a.s.--finite limit infimum. However, if $n \geq n_c$, then there is an a.s.--finite time at which the set consists of clusters of between $m+1$ and $2m+1$ elements, and forever after which these clusters grow apart, resulting in unchecked diameter growth. The existence of critical numerosities means that collectives can exhibit "phase transitions" that are governed purely by their numerosity and not, for example, their density or the strength of their interactions. This fact challenges prevalent beliefs about collective behavior and suggests new functionality for programmable matter. More broadly, it demonstrates an opportunity to explore the possible behaviors collectives through the study of random processes that rearrange sets.

math.PR↗

arXiv2021

Optimal discharge of patients from intensive care via a data-driven policy learning framework

Clinical decision support tools rooted in machine learning and optimization can provide significant value to healthcare providers, including through better management of intensive care units. In particular, it is important that the patient discharge task addresses the nuanced trade-off between decreasing a patient's length of stay (and associated hospitalization costs) and the risk of readmission or even death following the discharge decision. This work introduces an end-to-end general framework for capturing this trade-off to recommend optimal discharge timing decisions given a patient's electronic health records. A data-driven approach is used to derive a parsimonious, discrete state space representation that captures a patient's physiological condition. Based on this model and a given cost function, an infinite-horizon discounted Markov decision process is formulated and solved numerically to compute an optimal discharge policy, whose value is assessed using off-policy evaluation strategies. Extensive numerical experiments are performed to validate the proposed framework using real-life intensive care unit patient data.

cs.LG↗

arXiv2021

Existence of a Phase Transition in Harmonic Activation and Transport

Harmonic activation and transport (HAT) is a stochastic process that rearranges finite subsets of $\mathbb{Z}^d$, one element at a time. Given a finite set $U \subset \mathbb{Z}^d$ with at least two elements, HAT removes $x$ from $U$ according to the harmonic measure of $x$ in $U$, and then adds $y$ according to the probability that simple random walk from $x$, conditioned to hit the remaining set, steps from $y$ when it first does so. In particular, HAT conserves the number of elements in $U$. We study the classification of HAT as recurrent or transient, as the dimension $d$ and number of elements $n$ in the initial set vary. It was recently proved that the stationary distribution of HAT (on sets viewed up to translation) exists when $d = 2$, for every number of elements $n \geq 2$. We prove that HAT exhibits a phase transition in both $d$ and $n$, in the sense that HAT is transient when $d \geq 5$ and $n \geq 4$. Remarkably, transience occurs in only one "way": The set splits into clusters of two or three elements, which then grow steadily, indefinitely separated. We call these clusters dimers and trimers. Underlying this characterization of transience is the fact that, from any set, HAT reaches a set consisting exclusively of dimers and trimers, in a number of steps and with at least a probability which depend on $d$ and $n$ only.

math.PR↗

arXiv2021

Collapse and Diffusion in Harmonic Activation and Transport

For an $n$-element subset $U$ of $\mathbb{Z}^2$, select $x$ from $U$ according to harmonic measure from infinity, remove $x$ from $U$, and start a random walk from $x$. If the walk leaves from $y$ when it first enters $U$, add $y$ to $U$. Iterating this procedure constitutes the process we call Harmonic Activation and Transport (HAT). HAT exhibits a phenomenon we refer to as collapse: informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion. To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among $n$-element subsets of $\mathbb{Z}^2$, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, $d$? Concerning the former, examples abound for which the harmonic measure is exponentially small in $n$. We prove that it can be no smaller than exponential in $n \log n$. Regarding the latter, the escape probability is at most the reciprocal of $\log d$, up to a constant factor. We prove it is always at least this much, up to an $n$-dependent factor.

math.PR↗

arXiv2019

Brownian structure in the KPZ fixed point

Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds. When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be embedded via the Robinson-Schensted-Knuth correspondence as the uppermost curve in an $\mathbb{N}$-indexed system of random continuous curves, the Airy line ensemble. Among our principal results is the assertion that the Airy$_2$ process enjoys a very strong similarity to Brownian motion $B$ (of rate two) on unit-order intervals; as a consequence, the Radon-Nikodym derivative of the law of $\mathcal{A}$ on say $[-1,1]$, with respect to the law of $B$ on this interval, lies in every $L^p$ space for $p \in (1,\infty)$. Our technique of proof harnesses a probabilistic resampling or {\em Brownian Gibbs} property satisfied by the Airy line ensemble after parabolic shift, and this article develops Brownian Gibbs analysis of this ensemble begun in [CH14] and pursued in [Ham19a]. Our Brownian comparison for scaled interface profiles is an element in the ongoing programme of studying KPZ universality via probabilistic and geometric methods of proof, aided by limited but essential use of integrable inputs. Indeed, the comparison result is a useful tool for studying this universality class. We present and prove several applications, concerning for example the structure of near ground states in Brownian last passage percolation, or Brownian structure in scaled interface profiles that arise from evolution from any element in a very general class of initial data.

math.PR↗

arXiv2018

Unifying particle-based and continuum models of hillslope evolution with a probabilistic scaling technique

Relationships between sediment flux and geomorphic processes are combined with statements of mass conservation, in order to create continuum models of hillslope evolution. These models have parameters which can be calibrated using available topographical data. This contrasts the use of particle-based models, which may be more difficult to calibrate, but are simpler, easier to implement, and have the potential to provide insight into the statistics of grain motion. The realms of individual particles and the continuum, while disparate in geomorphological modeling, can be connected using scaling techniques commonly employed in probability theory. Here, we motivate the choice of a particle-based model of hillslope evolution, whose stationary distributions we characterize. We then provide a heuristic scaling argument, which identifies a candidate for their continuum limit. By simulating instances of the particle model, we obtain equilibrium hillslope profiles and probe their response to perturbations. These results provide a proof-of-concept in the unification of microscopic and macroscopic descriptions of hillslope evolution through probabilistic techniques, and simplify the study of hillslope response to external influences.

physics.geo-ph↗