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Jaron Kent-Dobias

Publications and source records attributed to Jaron Kent-Dobias.

At least 19 recordsLinked to original sources
arXiv2026

Stationary point complexity via minimal supersymmetry breaking

The statistics of stationary points are a powerful way to understand mean-field random landscapes, and the Kac--Rice formula is a general way to compute them. A longstanding technical barrier to these calculations is the presence of the absolute value of the determinant of the Hessian. Neglecting the absolute value produces an elegant 2-index supersymmetric representation of the problem, but is often incorrect. We develop an expanded 4-index supersymmetric representation of the complexity problem which incorporates the absolute value naturally via spontaneous supersymmetry breaking along a particular superspace direction. Positing that no additional symmetry breaking occurs implies the reduction to five order parameters corresponding to elements of a superspace operator algebra generated by the spontaneously SUSY-breaking operator. We relate the order parameters to the geometry and spectra of stationary points, showing that the SUSY-breaking order parameter corresponds to the spectral density of the Hessian at zero eigenvalue. We give examples of this formalism applied to calculate the annealed complexity of several models, including the perceptron and the Sherrington--Kirkpatrick model. The framework is naturally extended to quenched complexity, where each order parameter corresponds to a replica matrix.

cond-mat.dis-nn↗
arXiv2026

Very persistent random walkers reveal transitions in landscape topology

We study the typical behavior of random walkers on the microcanonical configuration space of mean-field disordered systems. Passive walks have an ergodicity-breaking transition at precisely the energy density associated with the dynamical glass transition, but persistent walks remain ergodic at lower energies. In models where the energy landscape is thoroughly understood, we show that, in the limit of infinite persistence time, the ergodicity-breaking transition coincides with a transition in the topology of microcanonical configuration space. We conjecture that this correspondence generalizes to other models, and use it to determine the topological transition energy in situations where the landscape properties are ambiguous.

cond-mat.dis-nn↗
arXiv2026

Structure of solutions to continuous constraint satisfaction problems through the statistics of wedged and inscribed spheres

The study of random landscapes has long relied on counting stationary points: metastable states and the barriers between them. However, this method is useless for describing flat regions, common in constraint satisfaction problems. We introduce a characterization of flat regions by counting the number of spheres that can be uniquely inserted into them, either by wedging spheres of fixed radius or by inscribing spheres of variable radius. The ratio of these counts constrains the topology of the solution space. We apply this characterization to the spherical perceptron and show the existence of at least two topological regimes.

cond-mat.dis-nn↗
arXiv2025

On the topology of solutions to random continuous constraint satisfaction problems

We consider the set of solutions to $M$ random polynomial equations whose $N$ variables are restricted to the $(N-1)$-sphere. Each equation has independent Gaussian coefficients and a target value $V_0$. When solutions exist, they form a manifold. We compute the average Euler characteristic of this manifold in the limit of large $N$, and find different behavior depending on the target value $V_0$, the ratio $α=M/N$, and the variances of the coefficients. We divide this behavior into five phases with different implications for the topology of the solution manifold. When $M=1$ there is a correspondence between this problem and level sets of the energy in the spherical spin glasses. We conjecture that the transition energy dividing two of the topological phases corresponds to the energy asymptotically reached by gradient descent from a random initial condition, possibly resolving an open problem in out-of-equilibrium dynamics. However, the quality of the available data leaves the question open for now.

cond-mat.dis-nn↗
arXiv2024

Conditioning the complexity of random landscapes on marginal optima

Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors are a vanishing minority compared with nonmarginal optima and other unstable stationary points. We introduce a generic technique for conditioning the statistics of stationary points in random landscapes on their marginality, and apply it in three isotropic settings with qualitatively different structure: in the spherical spin-glasses, where the energy is Gaussian and its Hessian is GOE; in multispherical spin glasses, which are Gaussian but non-GOE; and in sums of squared spherical random functions, which are non-Gaussian. In these problems we are able to fully characterize the distribution of marginal optima in the landscape, including when they are in the minority.

cond-mat.dis-nn↗
arXiv2024

Algorithm-independent bounds on complex optimization through the statistics of marginal optima

Optimization seeks extremal points in a function. When there are superextensively many optima, optimization algorithms are liable to get stuck. Under these conditions, generic algorithms tend to find marginal optima, which have many nearly flat directions. In a companion paper, we introduce a technique to count marginal optima in random landscapes. Here, we use the statistics of marginal optima calculated using this technique to produce generic bounds on optimization, based on the simple principle that algorithms will overwhelmingly tend to get stuck only where marginal optima are found. We demonstrate the idea on a simple non-Gaussian problem of maximizing the sum of squared random functions on a compact space. Numeric experiments using both gradient descent and generalized approximate message passing algorithms fall inside the expected bounds.

cond-mat.dis-nn↗
arXiv2024

Origin of Symmetry Breaking in the Grasshopper Model

The planar grasshopper problem, originally introduced in (Goulko & Kent 2017 Proc. R. Soc. A 473, 20170494), is a striking example of a model with long-range isotropic interactions whose ground states break rotational symmetry. In this work we analyze and explain the nature of this symmetry breaking with emphasis on the importance of dimensionality. Interestingly, rotational symmetry is recovered in three dimensions for small jumps, which correspond to the non-isotropic cogwheel regime of the two-dimensional problem. We discuss simplified models that reproduce the symmetry properties of the original system in N dimensions. For the full grasshopper model in two dimensions we obtain quantitative predictions for optimal perturbations of the disk. Our analytical results are confirmed by numerical simulations.

math-ph↗
arXiv2023

Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

The mixed spherical models were recently found to violate long-held assumptions about mean-field glassy dynamics. In particular, the threshold energy, where most stationary points are marginal and that in the simpler pure models attracts long-time dynamics, seems to lose significance. Here, we compute the typical distribution of stationary points relative to each other in mixed models with a replica symmetric complexity. We examine the stability of nearby points, accounting for the presence of an isolated eigenvalue in their spectrum due to their proximity. Despite finding rich structure not present in the pure models, we find nothing that distinguishes the points that do attract the dynamics. Instead, we find new geometric significance of the old threshold energy, and invalidate pictures of the arrangement of most marginal inherent states into a continuous manifold.

cond-mat.dis-nn↗
arXiv2023

When is the average number of saddle points typical?

A common measure of a function's complexity is the count of its stationary points. For complicated functions, this count grows exponentially with the volume and dimension of their domain. In practice, the count is averaged over a class of functions (the annealed average), but the large numbers involved can produce averages biased by extremely rare samples. Typical counts are reliably found by taking the average of the logarithm (the quenched average), which is more difficult and not often done in practice. When most stationary points are uncorrelated with each other, quenched and anneals averages are equal. Equilibrium heuristics can guarantee when most of the lowest minima will be uncorrelated. We show that these equilibrium heuristics cannot be used to draw conclusions about other minima and saddles by producing examples among Gaussian-correlated functions on the hypersphere where the count of certain saddles and minima has different quenched and annealed averages, despite being guaranteed `safe' in the equilibrium setting. We determine conditions for the emergence of nontrivial correlations between saddles, and discuss the implications for the geometry of those functions and what out-of-equilibrium settings might be affected.

cond-mat.stat-mech↗
arXiv2023

Normal forms, universal scaling functions, and extending the validity of the RG

Our community has a deep and sophisticated understanding of phase transitions and their universal scaling functions. We outline and advocate an ambitious program to use this understanding as an anchor for describing the surrounding phases. We explain how to use normal form theory to write universal scaling functions in systems where the renormalization-group flows cannot be linearized. We use the 2d Ising model to demonstrate how to calculate high-precision implementations of universal scaling functions, and how to extend them into a complete description of the surrounding phases. We discuss prospects and challenges involved into extending these early successes to the many other systems where the RG has successfully described emergent scale invariance, making them invaluable tools for engineers, biologists, and social scientists studying complex systems.

cond-mat.stat-mech↗
arXiv2022

How to count in hierarchical landscapes: a 'full' solution to mean-field complexity

We derive the general solution for counting the stationary points of mean-field complex landscapes. It incorporates Parisi's solution for the ground state, as it should. Using this solution, we count the stationary points of two models: one with multi-step replica symmetry breaking, and one with full replica symmetry breaking.

cond-mat.stat-mech↗
arXiv2022

Analytic continuation over complex landscapes

In this paper we follow up the study of 'complex complex landscapes,' rugged landscapes of many complex variables. Unlike real landscapes, the classification of saddles by index is trivial. Instead, the spectrum of fluctuations at stationary points determines their topological stability under analytic continuation of the theory. Topological changes, which occur at so-called Stokes points, proliferate among saddles with marginal (flat) directions and are suppressed otherwise. This gives a direct interpretation of the gap or 'threshold' energy -- which in the real case separates saddles from minima -- as the level where the spectrum of the hessian matrix of stationary points develops a gap. This leads to different consequences for the analytic continuation of real landscapes with different structures: the global minima of 'one step replica-symmetry broken' landscapes lie beyond a threshold, their hessians are gapped, and are locally protected from Stokes points, whereas those of 'many step replica-symmetry broken' have gapless hessians and Stokes points immediately proliferate. A new matrix ensemble is found, playing the role that GOE plays for real landscapes in determining the topological nature of saddles.

cond-mat.stat-mech↗
arXiv2021

Smooth and global Ising universal scaling functions

We describe a method for approximating the universal scaling functions for the Ising model in a field. By making use of parametric coordinates, the free energy scaling function has a polynomial series everywhere. Its form is taken to be a sum of the simplest functions that contain the singularities which must be present: the Langer essential singularity and the Yang--Lee edge singularity. Requiring that the function match series expansions in the low- and high-temperature zero-field limits fixes the parametric coordinate transformation. For the two-dimensional Ising model, we show that this procedure converges exponentially with the order to which the series are matched, up to seven digits of accuracy. To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy.

cond-mat.stat-mech↗
arXiv2020

Complex complex landscapes

We study the saddle-points of the $p$-spin model -- the best understood example of a `complex' (rugged) landscape -- when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal N}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the Bézout bound $\log\overline{\mathcal N}\sim N\log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square-root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.

cond-mat.stat-mech↗
arXiv2020

Elastic properties of hidden order in URu$_{\text2}$Si$_{\text2}$ are reproduced by a staggered nematic

We develop a phenomenological mean field theory describing the hidden order phase in URu$_{\text2}$Si$_{\text2}$ as a nematic of the $\text B_{\text{1g}}$ representation staggered along the $c$-axis. Several experimental features are reproduced by this theory: the topology of the temperature--pressure phase diagram, the response of the elastic modulus $(C_{11}-C_{12})/2$ above the transition at ambient pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, hidden order is characterized by broken rotational symmetry that is modulated along the $c$-axis, the primary order of the high-pressure phase is an unmodulated nematic, and the triple point joining those two phases with the high-temperature paramagnetic phase is a Lifshitz point.

cond-mat.str-el↗
arXiv2020

Visualizing probabilistic models in Minkowski space with intensive symmetrized Kullback-Leibler embedding

We show that the predicted probability distributions for any $N$-parameter statistical model taking the form of an exponential family can be explicitly and analytically embedded isometrically in a $N{+}N$-dimensional Minkowski space. That is, the model predictions can be visualized as control parameters are varied, preserving the natural distance between probability distributions. All pairwise distances between model instances are given by the symmetrized Kullback-Leibler divergence. We give formulas for these intensive symmetrized Kullback Leibler (isKL) coordinate embeddings, and illustrate the resulting visualizations with the Bernoulli (coin toss) problem, the ideal gas, $n$ sided die, the nonlinear least squares fit, and the Gaussian fit. We highlight how isKL can be used to determine the minimum number of parameters needed to describe probabilistic data, and conclude by visualizing the prediction space of the two-dimensional Ising model, where we examine the manifold behavior near its critical point.

cond-mat.stat-mech↗
arXiv2018

Cluster representations and the Wolff algorithm in arbitrary external fields

We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, like Wolff, to systems in arbitrary fields, be they linear magnetic vector fields or nonlinear anisotropic ones. By generalizing the 'ghost spin' representation to one with a 'ghost transformation,' global invariance to spin symmetry transformations is restored at the cost of an extra degree of freedom which lives in the space of symmetry transformations. The ordinary cluster-building process can then be run on the new representation. We show that this extension preserves the scaling of accelerated dynamics in the absence of a field for Ising, Potts, and $\mathrm O(n)$ models and demonstrate the method's use in modelling the presence of novel nonlinear fields. We also provide a C++ library for the method's convenient implementation for arbitrary models.

cond-mat.stat-mech↗