Cost of Diffusion: Nonlinearity and Giant Fluctuations.

We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model-inspired by the way taxi meters work-exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance traveled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting "freezing" transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force.

Instabilities of complex fluids with partially structured and partially random interactions.

We develop a theory for thermodynamic instabilities of complex fluids composed of many interacting chemical species organised in families. This model includes partially structured and partially random interactions and can be solved exactly using tools from random matrix theory. The model exhibits three kinds of fluid instabilities: one in which the species form a condensate with a local density that depends on their family (family condensation); one in which species demix in two phases depending on their family (family demixing); and one in which species demix in a random manner irrespective of their family (random demixing). We determine the critical spinodal density of the three types of instabilities and find that the critical spinodal density is finite for both family condensation and family demixing, while for random demixing the critical spinodal density grows as the square root of the number of species. We use the developed framework to describe phase-separation instability of the cytoplasm induced by a change in pH.

Universal covariance formula for linear statistics on random matrices.

We derive an analytical formula for the covariance cov(A,B) of two smooth linear statistics A=[under ∑]ia(λ_{i}) and B=[under ∑]ib(λ_{i}) to leading order for N→∞, where {λ_{i}} are the N real eigenvalues of a general one-cut random-matrix model with Dyson index ß. The formula, carrying the universal 1/ß prefactor, depends on the random-matrix ensemble only through the edge points [λ_{-},λ_{+}] of the limiting spectral density. For A=B, we recover in some special cases the classical variance formulas by Beenakker and by Dyson and Mehta, clarifying the respective ranges of applicability. Some choices of a(x) and b(x) lead to a striking decorrelation of the corresponding linear statistics. We provide two applications-the joint statistics of conductance and shot noise in ideal chaotic cavities, and some new fluctuation relations for traces of powers of random matrices.

Phase transitions and edge scaling of number variance in Gaussian random matrices.

We consider N × N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-√2],√2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P(N,L)(N(L)) that N(L) eigenvalues lie within the box [-L,L]. This probability scales as P(N,L)(N(L) = κ(L)N) ≈ exp(-ßN(2)ψ(L)(κ(L))), where ß is the Dyson index of the ensemble and ψ(L)(κ(L)) is a ß-independent rate function that we compute exactly. We identify three regimes as L is varied: (i) N(-1)≪L < √2 (bulk), (ii) L∼√2 on a scale of O(N(-2/3)) (edge), and (iii) L > sqrt[2] (tail). We find a dramatic nonmonotonic behavior of the number variance V(N)(L) as a function of L: after a logarithmic growth ∝ln(NL) in the bulk (when L∼O(1/N)), V(N)(L) decreases abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > sqrt[2]. This "dropoff" of V(N)(L) at the edge is described by a scaling function V(ß) that smoothly interpolates between the bulk (i) and the tail (iii). For ß = 2 we compute V(2) explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for ß = 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap.

Delocalization-localization dynamical phase transition of random walks on graphs.

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a "coexistence" of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite-size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

Nonlinear-cost random walk: Exact statistics of the distance covered for fixed budget.

We consider the nonlinear-cost random-walk model in discrete time introduced in Phys. Rev. Lett. 130, 237102 (2023)10.1103/PhysRevLett.130.237102, where a fee is charged for each jump of the walker. The nonlinear cost function is such that slow or short jumps incur a flat fee, while for fast or long jumps the cost is proportional to the distance covered. In this paper we compute analytically the average and variance of the distance covered in n steps when the total budget C is fixed, as well as the statistics of the number of long or short jumps in a trajectory of length n, for the exponential jump distribution. These observables exhibit a very rich and nonmonotonic scaling behavior as a function of the variable C/n, which is traced back to the makeup of a typical trajectory in terms of long or short jumps, and the resulting entropy thereof. As a by-product, we compute the asymptotic behavior of ratios of Kummer hypergeometric functions when both the first and last arguments are large. All our analytical results are corroborated by numerical simulations.

Graphie: A network-based visual interface for the UK's primary legislation.

Background: legislation.gov.uk is a platform that enables users to explore and navigate the many sections of the UK's legal corpus through its well-designed searching and browsing features. However, there is room for improvement as it lacks the ability to easily move between related sections or Acts and only presents a text-only rendering of provisions. With Graphie, our novel navigational tool (graphie.quantlaw.co.uk), we aim to address this limitation by presenting alternative visualizations of legal documents using both text and graphs. Methods: The building block of Graphie is Sofia, an offline data pipeline designed to support different data visualizations by parsing and modelling data provided by legislation.gov.uk in open access form. Results: Graphie provides a network representation of the hierarchical structure of an Act of Parliament, which is typically organized in a tree-like fashion according to the content and information contained in each sub-branch. Nodes in Graphie represent sections of an Act (or individual provisions), while links embody the hierarchical connections between them. The legal map provided by Graphie is easily navigable by hovering on nodes, which are also color-coded and numbered to provide easily accessible information about the underlying content. The full textual content of each node is also available on a dedicated hyperlinked canvas. Conclusions: While we focus on the Housing Act 2004 for illustrative purposes, our platform is scalable, versatile, and provides users with a unified toolbox to visualize and explore the UK legal corpus in a fast and user-friendly way.

Number of relevant directions in principal component analysis and Wishart random matrices.

We compute analytically, for large N, the probability P(N+,N) that a N×N Wishart random matrix has N+ eigenvalues exceeding a threshold Nζ, including its large deviation tails. This probability plays a benchmark role when performing the principal component analysis of a large empirical data set. We find that P(N+,N)≈exp[-ßN2ψζ(N+/N)], where ß is the Dyson index of the ensemble and ψζ(κ) is a rate function that we compute explicitly in the full range 0≤κ≤1 and for any ζ. The rate function ψζ(κ) displays a quadratic behavior modulated by a logarithmic singularity close to its minimum κ⋆(ζ). This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance Δ(N) of the number of relevant components is also shown to grow universally (independent of ζ) as Δ(N)∼(ßπ2)-1 lnN for large N.

Equilibrium strategy and population-size effects in lowest unique bid auctions.

In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large data set of internet auctions. The theory agrees with the data with striking accuracy for small population-size N, while for larger N a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.

Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with multiple transverse modes.

We compute analytically the full distribution of Andreev conductance G(NS) of a metal-superconductor interface with a large number N(c) of transverse modes, using a random matrix approach. The probability distribution P(G(NS),N(c) in the limit of large N(c) displays a Gaussian behavior near the average value =(2-√2)N(c) and asymmetric power-law tails in the two limits of very small and very large G(NS). In addition, we find a novel third regime sandwiched between the central Gaussian peak and the power-law tail for large G(NS). Weakly nonanalytic points separate these four regimes-these are shown to be consequences of three phase transitions in an associated Coulomb gas problem.

Maximal modularity and the optimal size of parliaments.

An important question in representative democracies is how to determine the optimal parliament size of a given country. According to an old conjecture, known as the cubic root law, there is a fairly universal power-law relation, with an exponent equal to 1/3, between the size of an elected parliament and the country's population. Empirical data in modern European countries support such universality but are consistent with a larger exponent. In this work, we analyse this intriguing regularity using tools from complex networks theory. We model the population of a democratic country as a random network, drawn from a growth model, where each node is assigned a constituency membership sampled from an available set of size D. We calculate analytically the modularity of the population and find that its functional relation with the number of constituencies is strongly non-monotonic, exhibiting a maximum that depends on the population size. The criterion of maximal modularity allows us to predict that the number of representatives should scale as a power-law in the size of the population, a finding that is qualitatively confirmed by the empirical analysis of real-world data.

A percolation model for the emergence of the Bitcoin Lightning Network.

The Lightning Network is a so-called second-layer technology built on top of the Bitcoin blockchain to provide "off-chain" fast payment channels between users, which means that not all transactions are settled and stored on the main blockchain. In this paper, we model the emergence of the Lightning Network as a (bond) percolation process and we explore how the distributional properties of the volume and size of transactions per user may impact its feasibility. The agents are all able to reciprocally transfer Bitcoins using the main blockchain and also - if economically convenient - to open a channel on the Lightning Network and transact "off chain". We base our approach on fitness-dependent network models: as in real life, a Lightning channel is opened with a probability that depends on the "fitness" of the concurring nodes, which in turn depends on wealth and volume of transactions. The emergence of a connected component is studied numerically and analytically as a function of the parameters, and the phase transition separating regions in the phase space where the Lightning Network is sustainable or not is elucidated. We characterize the phase diagram determining the minimal volume of transactions that would make the Lightning Network sustainable for a given level of fees or, alternatively, the maximal cost the Lightning ecosystem may impose for a given average volume of transactions. The model includes parameters that could be in principle estimated from publicly available data once the evolution of the Lighting Network will have reached a stationary operable state, and is fairly robust against different choices of the distributions of parameters and fitness kernels.

Publisher's Note: Phase transitions in the condition-number distribution of Gaussian random matrices [Phys. Rev. E 90, 050103(R) (2014)].

This corrects the article DOI: 10.1103/PhysRevE.90.050103.

A minimalistic model of bias, polarization and misinformation in social networks.

Online social networks provide users with unprecedented opportunities to engage with diverse opinions. At the same time, they enable confirmation bias on large scales by empowering individuals to self-select narratives they want to be exposed to. A precise understanding of such tradeoffs is still largely missing. We introduce a social learning model where most participants in a network update their beliefs unbiasedly based on new information, while a minority of participants reject information that is incongruent with their preexisting beliefs. This simple mechanism generates permanent opinion polarization and cascade dynamics, and accounts for the aforementioned tradeoff between confirmation bias and social connectivity through analytic results. We investigate the model's predictions empirically using US county-level data on the impact of Internet access on the formation of beliefs about global warming. We conclude by discussing policy implications of our model, highlighting the downsides of debunking and suggesting alternative strategies to contrast misinformation.

Random matrix approach to collective behavior and bulk universality in protein dynamics.

Covariance matrices of amino acid displacements, commonly used to characterize the large-scale movements of proteins, are investigated through the prism of random matrix theory. Bulk universality is detected in the local spacing statistics of noise-dressed eigenmodes, which is well described by a Brody distribution with parameter beta approximately = 0.8. This finding, supported by other consistent indicators, implies a novel quantitative criterion to single out the collective degrees of freedom of the protein from the majority of high-energy, localized vibrations.

Index distribution of gaussian random matrices.

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N+) of a random N x N matrix belonging to Gaussian orthogonal (beta=1), unitary (beta=2) or symplectic (beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N+/N scales, for large N, as P(c,N) approximately = exp[-betaN(2)Phi(c)] where the rate function Phi(c), symmetric around c=1/2 and universal (independent of beta), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.

Exact distributions of currents and frenesy for Markov bridges.

We consider discrete-time Markov bridges, chains whose initial and final states coincide. We derive exact finite-time formulae for the joint probability distributions of additive functionals of trajectories. We apply our theory to time-integrated currents and frenesy of enzymatic reactions, which may include absolutely irreversible transitions. We discuss the information that frenesy carries about the currents and show that bridges may violate known uncertainty relations in certain cases. Numerical simulations are in perfect agreement with our theory.

Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls.

Consider the free energy of a d-dimensional gas in canonical equilibrium under pairwise repulsive interaction and global confinement, in presence of a volume constraint. When the volume of the gas is forced away from its typical value, the system undergoes a phase transition of the third order separating two phases (pulled and pushed). We prove this result (i) for the eigenvalues of one-cut, off-critical random matrices (log-gas in dimension d = 1 ) with hard walls; (ii) in arbitrary dimension d ≥ 1 for a gas with Yukawa interaction (aka screened Coulomb gas) in a generic confining potential. The latter class includes systems with Coulomb (long range) and delta (zero range) repulsion as limiting cases. In both cases, we obtain an exact formula for the free energy of the constrained gas which explicitly exhibits a jump in the third derivative, and we identify the 'electrostatic pressure' as the order parameter of the transition. Part of these results were announced in Cunden et al. (J Phys A 51:35LT01, 2018).

Eigenvector statistics of the product of Ginibre matrices.

We develop a method to calculate left-right eigenvector correlations of the product of m independent N×N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O=1+(m/2)(N-1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N→∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.