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.

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.

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.

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.

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.

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.

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.

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.

Number statistics for ß-ensembles of random matrices: Applications to trapped fermions at zero temperature.

Let P_{ß}^{(V)}(N_{I}) be the probability that a N×Nß-ensemble of random matrices with confining potential V(x) has N_{I} eigenvalues inside an interval I=[a,b] on the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically P_{ß}^{(V)}(N_{I}) for large N. We show that this probability scales for large N as P_{ß}^{(V)}(N_{I})≈exp[-ßN^{2}ψ^{(V)}(N_{I}/N)], where ß is the Dyson index of the ensemble. The rate function ψ^{(V)}(k_{I}), independent of ß, is computed in terms of single integrals that can be easily evaluated numerically. The general formalism is then applied to the classical ß-Gaussian (I=[-L,L]), ß-Wishart (I=[1,L]), and ß-Cauchy (I=[-L,L]) ensembles. Expanding the rate function around its minimum, we find that generically the number variance var(N_{I}) exhibits a nonmonotonic behavior as a function of the size of the interval, with a maximum that can be precisely characterized. These analytical results, corroborated by numerical simulations, provide the full counting statistics of many systems where random matrix models apply. In particular, we present results for the full counting statistics of zero-temperature one-dimensional spinless fermions in a harmonic trap.

Random pure states: Quantifying bipartite entanglement beyond the linear statistics.

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

Phase transitions in the condition-number distribution of Gaussian random matrices.

We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and smallest squared singular values) of N×M Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large N the cumulative P(κx) distributions of κ. We find that these distributions decay as P(κx)≈exp[-ßNΦ_{+}(x)], where ß is the Dyson index of the ensemble. The left and right rate functions Φ_{±}(x) are independent of ß and calculated exactly for any choice of the rectangularity parameter α=M/N-1>0. Interestingly, they show a weak nonanalytic behavior at their minimum 〈κ〉 (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around 〈κ〉, we determine exactly the scale of typical fluctuations ∼O(N^{-2/3}) and the tails of the limiting distribution of κ. The analytical results are in excellent agreement with numerical simulations.

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.

Singular-potential random-matrix model arising in mean-field glassy systems.

We consider an invariant random matrix ensemble where the standard Gaussian potential is distorted by an additional single pole of arbitrary fixed order. Potentials with first- and second-order poles have been considered previously and found applications in quantum chaos and number theory. Here we present an application to mean-field glassy systems. We derive and solve the loop equation in the planar limit for the corresponding class of potentials. We find that the resulting mean or macroscopic spectral density is generally supported on two disconnected intervals lying on the two sides of the repulsive pole, whose edge points can be completely determined imposing the additional constraint of traceless matrices on average. For an unbounded potential with an attractive pole, we also find a possible one-cut solution for certain values of the couplings, which is ruled out when the traceless condition is imposed. Motivated by the calculation of the distribution of the spin-glass susceptibility in the Sherrington-Kirkpatrick spin-glass model, we consider in detail a second-order pole for a zero-trace model and provide the most explicit solution in this case. In the limit of a vanishing pole, we recover the standard semicircle. Working in the planar limit, our results apply to matrices with orthogonal, unitary, and symplectic invariance. Numerical simulations and an independent analytical Coulomb fluid calculation for symmetric potentials provide an excellent confirmation of our results.

Statistics of quantum transport in weakly nonideal chaotic cavities.

We consider statistics of electronic transport in chaotic cavities where time-reversal symmetry is broken and one of the leads is weakly nonideal; that is, it contains tunnel barriers characterized by tunneling probabilities Γ(i). Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbation theory in 1-Γ(i) valid for an arbitrary number of channels and obtain explicit formulas up to second order for the average and variance of the conductance and for the average shot noise. Higher moments of the conductance are considered to leading order.