arXiv subjects

Dana Randall

Publications and source records attributed to Dana Randall.

At least 19 recordsLinked to original sources

arXiv2026

Slow Convergence of Ising and Spin Glass Models with Well-Separated Frustrated Vertices

The ferromagnetic Ising model on an $n\times n$ square lattice region $\Lambda$ with mixed boundary conditions can exhibit a phase transition as temperature varies. For this spin system, if we fix the spins on the top and bottom sides of the square to be $+$ and the left and right sides to be~$-$, a standard Peierls argument based on energy shows that below some critical temperature~$t_c$, any local Markov chain $\mathcal{M}$ requires time exponential in $n$ to mix. Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered {\it frustrated} because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under $90$ degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This argument extends to the ferromagnetic Ising model with mixed boundary conditions described above, which behaves like spin glasses with frustrated faces on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of $\Lambda$ and yields weaker results when they are on the boundary of $\Lambda$ but not near the corners. We show that there is a universal temperature $T$ below which $\mathcal{M}$ will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the {\it free energy}, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.

cs.DM↗

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

Splittable Spanning Trees and Balanced Forests in Dense Random Graphs

We consider the probability that a spanning tree chosen uniformly at random from a graph can be partitioned into a fixed number $k$ of trees of equal size by removing $k-1$ edges. In that case, the spanning tree is called {\em splittable}. Splittable spanning trees are useful in algorithms for sampling {\em balanced forests}, forests whose components are of equal size, and for sampling partitions of a graph into components of equal size, with applications in redistricting, network algorithms, and image decomposition. Cannon et al.~recently showed that spanning trees on grid and grid-like graphs on $n$ vertices are splittable into $k$ equal sized components with probability at least $n^{-2k}$, leading to the first rigorous sampling algorithm for balanced forests in any class of graphs. Focusing on the complementary case of dense random graphs, we show that random spanning trees have inverse polynomial probability of being splittable; specifically, a random spanning tree is splittable with probability at least $n^{(-k/2)}$ for both the $G(n,p)$ and $G(n,m)$ models when $p = \Omega(1/\log n)$, giving the first dense class of graphs where partitions of equal size can be sampled efficiently. In addition, we present an infinite family of graphs with properties that have been conjectured to ensure splittability (i.e., Hamiltonian subgraphs of the triangular lattice) and prove that random spanning trees are not splittable with more than exponentially small probability. As a consequence, we show that a family of widely-used Markov chain algorithms for sampling equal-size partitions will fail on this family of graphs if their state spaces are restricted to equal-size partitions. Moreover, we show these algorithms will be inefficient if their state spaces are generalized to include any unbalanced partitions, suggesting barriers for sampling balanced partitions in sparse graphs.

cs.DS↗

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

Adaptive Collective Responses to Local Stimuli in Anonymous Dynamic Networks

We develop a framework for self-induced phase changes in programmable matter in which a collection of agents with limited computational and communication capabilities can collectively perform appropriate global tasks in response to local stimuli that dynamically appear and disappear. Agents reside on graph vertices, where each stimulus is only recognized locally, and agents communicate via token passing along edges to alert other agents to transition to an "aware" state when stimuli are present and an "unaware" state when the stimuli disappear. We present an Adaptive Stimuli Algorithm that is robust to competing waves of messages as multiple stimuli change, possibly adversarially. Moreover, in addition to handling arbitrary stimulus dynamics, the algorithm can handle agents reconfiguring the connections (edges) of the graph over time in a controlled way. As an application, we show how this Adaptive Stimuli Algorithm on reconfigurable graphs can be used to solve the foraging problem, where food sources may be discovered, removed, or shifted at arbitrary times. We would like the agents to consistently self-organize using only local interactions, such that if the food remains in position long enough, the agents transition to a gather phase, collectively forming a single large component with small perimeter around the food. Alternatively, if no food source has existed recently, the agents should self-induce a switch to a search phase in which they distribute themselves randomly throughout the lattice region to search for food. Unlike previous approaches to foraging, this process is indefinitely repeatable. Like a physical phase change, microscopic changes such as the deletion or addition of a single food source triggers these macroscopic, system-wide transitions as agents share information about the environment and respond locally to get the desired collective response.

cs.DC↗

arXiv2022

A Heterogeneous Schelling Model for Wealth Disparity and its Effect on Segregation

The Schelling model of segregation was introduced in economics to show how micro-motives can influence macro-behavior. Agents on a lattice have two colors and try to move to a different location if the number of their neighbors with a different color exceeds some threshold. Simulations reveal that even such mild local color preferences, or homophily, are sufficient to cause segregation. In this work, we propose a stochastic generalization of the Schelling model, based on both race and wealth, to understand how carefully architected placement of incentives, such as urban infrastructure, might affect segregation. In our model, each agent is assigned one of two colors along with a label, rich or poor. Further, we designate certain vertices on the lattice as "urban sites," providing civic infrastructure that most benefits the poorer population, thus incentivizing the occupation of such vertices by poor agents of either color. We look at the stationary distribution of a Markov process reflecting these preferences to understand the long-term effects. We prove that when incentives are large enough, we will have "urbanization of poverty," an observed effect whereby poor people tend to congregate on urban sites. Moreover, even when homophily preferences are very small, if the incentives are large and there is income inequality in the two-color classes, we can get racial segregation on urban sites but integration on non-urban sites. In contrast, we find overall mitigation of segregation when the urban sites are distributed throughout the lattice and the incentives for urban sites exceed the homophily biases. We prove that in this case, no matter how strong homophily preferences are, it will be exponentially unlikely that a configuration chosen from stationarity will have large, homogeneous clusters of agents of either color, suggesting we will have racial integration with high probability.

nlin.AO↗

arXiv2022

Mathematically Quantifying Non-responsiveness of the 2021 Georgia Congressional Districting Plan

To audit political district maps for partisan gerrymandering, one may determine a baseline for the expected distribution of partisan outcomes by sampling an ensemble of maps. One approach to sampling is to use redistricting policy as a guide to precisely codify preferences between maps. Such preferences give rise to a probability distribution on the space of redistricting plans, and Metropolis-Hastings methods allow one to sample ensembles of maps from the specified distribution. Although these approaches have nice theoretical properties and have successfully detected gerrymandering in legal settings, sampling from commonly-used policy-driven distributions is often computationally difficult. As of yet, there is no algorithm that can be used off-the-shelf for checking maps under generic redistricting criteria. In this work, we mitigate the computational challenges in a Metropolized-sampling technique through a parallel tempering method combined with ReCom[12] and, for the first time, validate that such techniques are effective on these problems at the scale of statewide precinct graphs for more policy informed measures. We develop these improvements through the first case study of district plans in Georgia. Our analysis projects that any election in Georgia will reliably elect 9 Republicans and 5 Democrats under the enacted plan. This result is largely fixed even as public opinion shifts toward either party and the partisan outcome of the enacted plan does not respond to the will of the people. Only 0.12% of the $\sim$160K plans in our ensemble were similarly non-responsive.

cs.CY↗

arXiv2022

Spanning Trees and Redistricting: New Methods for Sampling and Validation

Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now signed on to decisions that find that computational methods can provide key evidence. Today, the leading approaches for benchmarking districting plans are based on the use of spanning trees for sampling graph partitions. We present a new *reversible recombination* algorithm and rigorously prove its fundamental properties. Furthermore, we argue for a canonical sampling distribution called the *spanning tree distribution* that is well adapted to redistricting and provides a principled foundation for comparing and validating methods. Together with a highly efficient (and open-source) implementation that can generate and handle large datasets, this work provides the most powerful null model to date for the gerrymandering problem, meeting an urgent democratic challenge with sound scientific methodology.

physics.soc-ph↗

arXiv2022

Local Stochastic Algorithms for Alignment in Self-Organizing Particle Systems

We present local distributed, stochastic algorithms for \emph{alignment} in self-organizing particle systems (SOPS) on two-dimensional lattices, where particles occupy unique sites on the lattice, and particles can make spatial moves to neighboring sites if they are unoccupied. Such models are abstractions of programmable matter, composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational capabilities. We consider oriented particle systems, where particles are assigned a vector pointing in one of $q$ directions, and each particle can compute the angle between its direction and the direction of any neighboring particle, although without knowledge of global orientation with respect to a fixed underlying coordinate system. Particles move stochastically, with each particle able to either modify its direction or make a local spatial move along a lattice edge during a move. We consider two settings: (a) where particle configurations must remain simply connected at all times and (b) where spatial moves are unconstrained and configurations can disconnect. Taking inspiration from the Potts and clock models from statistical physics, we prove that for any $q \geq 2,$ these self-organizing particle systems can be made to collectively align along a single dominant direction (analogous to a solid or ordered state) or remain non-aligned, in which case the fraction of particles oriented along any direction is nearly equal (analogous to a gaseous or disordered state). Moreover, we show that with appropriate settings of the input parameters, we can achieve \emph{compression} and \emph{expansion}, controlling how tightly gathered the particles are, as well as \emph{alignment} or \emph{nonalignment}, producing a single dominant orientation or not.

cs.DC↗

arXiv2022

Foraging in Particle Systems via Self-Induced Phase Changes

The foraging problem asks how a collective of particles with limited computational, communication and movement capabilities can autonomously compress around a food source and disperse when the food is depleted or shifted, which may occur at arbitrary times. We would like the particles to iteratively self-organize, using only local interactions, to correctly gather whenever a food particle remains in a position long enough and search if no food particle has existed recently. Unlike previous approaches, these search and gather phases should be self-induced so as to be indefinitely repeatable as the food evolves, with microscopic changes to the food triggering macroscopic, system-wide phase transitions. We present a stochastic foraging algorithm based on a phase change in the fixed magnetization Ising model from statistical physics: Our algorithm is the first to leverage self-induced phase changes as an algorithmic tool. A key component of our algorithm is a careful token passing mechanism ensuring a dispersion broadcast wave will always outpace a compression wave. We also present a highly structured alternative algorithm that gathers by incrementally building a spiral tightly wrapped around the food particle.

cs.DC↗

arXiv2021

Programming Active Cohesive Granular Matter with Mechanically Induced Phase Changes

Active matter physics and swarm robotics have provided powerful tools for the study and control of ensembles driven by internal sources. At the macroscale, controlling swarms typically utilizes significant memory, processing power, and coordination unavailable at the microscale, e.g., for colloidal robots, which could be useful for fighting disease, fabricating intelligent textiles, and designing nanocomputers. To develop principles that that can leverage physics of interactions and thus can be utilized across scales, we take a two-pronged approach: a theoretical abstraction of self-organizing particle systems and an experimental robot system of active cohesive granular matter that intentionally lacks digital electronic computation and communication, using minimal (or no) sensing and control, to test theoretical predictions. We consider the problems of aggregation, dispersion, and collective transport. As predicted by the theory, as a parameter representing interparticle attraction increases, the robots transition from a dispersed phase to an aggregated one, forming a dense, compact collective. When aggregated, the collective can transport non-robot "impurities" in their environment, thus performing an emergent task driven by the physics underlying the transition. These results point to a fruitful interplay between algorithm design and active matter robophysics that can result in new nonequilibrium physics and principles for programming collectives without the need for complex algorithms or capabilities.

cond-mat.soft↗

arXiv2020

Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

The six-vertex model in statistical physics is a weighted generalization of the ice model on $\mathbb{Z}^2$ (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks.

cs.DS↗

arXiv2019

Thermodynamic Computing

The hardware and software foundations laid in the first half of the 20th Century enabled the computing technologies that have transformed the world, but these foundations are now under siege. The current computing paradigm, which is the foundation of much of the current standards of living that we now enjoy, faces fundamental limitations that are evident from several perspectives. In terms of hardware, devices have become so small that we are struggling to eliminate the effects of thermodynamic fluctuations, which are unavoidable at the nanometer scale. In terms of software, our ability to imagine and program effective computational abstractions and implementations are clearly challenged in complex domains. In terms of systems, currently five percent of the power generated in the US is used to run computing systems - this astonishing figure is neither ecologically sustainable nor economically scalable. Economically, the cost of building next-generation semiconductor fabrication plants has soared past $10 billion. All of these difficulties - device scaling, software complexity, adaptability, energy consumption, and fabrication economics - indicate that the current computing paradigm has matured and that continued improvements along this path will be limited. If technological progress is to continue and corresponding social and economic benefits are to continue to accrue, computing must become much more capable, energy efficient, and affordable. We propose that progress in computing can continue under a united, physically grounded, computational paradigm centered on thermodynamics. Herein we propose a research agenda to extend these thermodynamic foundations into complex, non-equilibrium, self-organizing systems and apply them holistically to future computing systems that will harness nature's innate computational capacity. We call this type of computing "Thermodynamic Computing" or TC.

cs.CY↗