We study the statistics of rewired random regular graphs (RRGs) in a mixed ensemble, where the average number of triangles is controlled by the fugacity λ, while the number of vertices and the vertex degree are fixed. This model exhibits a phase transition at critical fugacity λ_cr from a triangle-poor phase (TPP), in which the number of triangles is independent of the system size, to a triangle-rich phase (TRP), in which the number of triangles scales linearly with the system size. We estimate λ_cr by comparing the entropy of TPP with the energy of TRP. Above λ_cr, the RRG becomes a two-phase system in which dense clusters are connected by a sparse scale-free sub-network characterized by a degree distribution, P(d) ∼ d^-γ, with γ≈ 2, independent of the size of the whole graph and its degree. We attribute this behavior to an "emergent preferential attachment" induced by triangle motifs, describe the mechanism underlying its formation, and derive the exponent γ within a mean-field approach. We show that most inter-cluster triangles are isosceles, with the base lying inside one cluster and the apex belonging to the inter-cluster network. Finally, we speculate on a possible connection between these triangles and Efimov states in a conformally invariant potential.

We show that continuous random walks (diffusion) in the Poincaré hyperbolic upper halfplane ℍ^2 = (x,y)|y>0, interpreted as multiplicative stochastic processes with log-normal statistics, provide a unifying framework linking three seemingly unrelated phenomena: (i) the non-analytic divergence of corrrelation length at the Berezinskii-Kosterlitz-Thouless (BKT) transition; (ii) the appearence of the Kardar-Parisi-Zhang 9KPZ) exponent in the fluctuational behavior of stretched random walks constrained above an impermeable disc; and (iii) the emergence of Lifshitz tails in one-dimensional statistics of rare events. Combining scaling arguments with analytic derivations and numerical analysis, we adapt the renormalization-group equations originally developed for the Efimov effect in a two-dimensional conformally invariant potential to the case of diffusion in ℍ^2, thereby deriving the BKT-type divergence of the correlation length. We further demonstrate how the KPZ-type scaling governs the large-deviation behavior and survival probability near the boundary in the hyperbolic domain, and how Lifshitz tails arise naturally in a deterministic large-deviation landscape on the hyperbolic plane via instanton approach, reproducing the rare-event statistics of one-dimensional diffusion in the array of traps with the Poisson distribution. We conjecture that the dominant contribution to the ensemble of paths responsible for BKT-like physics comes from random paths pushed to large-deviation stretched regime.

The N-step random walk, elongated in the vicinity of a disk (in 2D) or a sphere (in 3D) of radius R, demonstrates a nonalgebraic stretched exponential decay P_{N}∼exp(-constN^{1/3}) for the first-return probability P_{N} in the double-scaling limit N=L/a≫1,R/a≫1 conditioned that L/R=c=const. Stretching means that the length of the walk, L=Na (where a is the unit step length) satisfies the condition L=cR, where c>π and under first return we understand the radial first arrival to a boundary. Both analytic and numerical evidence of the nonalgebraic behavior of P_{N} are provided. Considering the model of a polymer loop stretched (inflated) by external force, we show that nonalgebraic behavior of P_{N} affects the adsorption of a polymer at the boundary of a sticky disk in 2D, manifesting in a first-order localization transition.

In this work, we explore the fundamental principles underlying the architecture of the human structural connectome. We use a simple generative model that combines nonlinear preferential attachment with biologically plausible geometric constraints. This model successfully reproduces a remarkable number of brain network properties, linking the overall architecture of the brain to its underlying evolutionary principles. Additionally, we investigate the localization properties of connectome eigenmodes and identify multiple indications of their fractality. Drawing on recent advances in Anderson localization in complex systems, we conjecture that the structural connectome operates in an extended critical phase that supports multifractality.

In our daily lives, we encounter numerous independent events, each occurring with varying probabilities over time. This work delves into the scientific background behind the inhomogeneous distribution of these events over time, often resulting in what we refer to as "black days", where multiple events seem to converge at once. In the first part of the work we performed an analysis involving D independent periodic and random sequences of events. Using the Uniform Manifold Approximation and Projection (UMAP) technique, we observed a clustering of event sequences on a two-dimensional manifold M at a certain large D. We interpret this clustering as a signature of "black days", which bears a clear resemblance to traffic jams in vehicle flow. In the second part of the work we examined in detail clustering patterns of independently distributed N points within the corners of a D-dimensional cube when 1 << N<D. Our findings revealed that a transition to a single-component cluster occurs at a critical dimensionality, D-cr, via a nearly third-order phase transition. We demonstrate that for large D, the number of disjoint components exhibits a "saw-tooth" pattern as a function of D. Analyzing the spectral density, rho(lambda), of the corresponding adjacency graph in the vicinity of the clustering transition we recover the singular "Lifshitz tail" behavior at the spectral boundary of rho(lambda).

Digitally connected societies approach a transparent regime where all agents can interact without geographic or social barriers – a limit realized by complete graph topologies. We solve exactly a q-state Potts model with many-body interactions on this geometry, modeling agents from q distinct communities. Analyzing the illustrative case of competing pairwise and three-body couplings, we identify three key phases in the thermodynamic limit: democratic (all communities equal), marginalized (q-1 communities surviving), and consensus (one dominant group). For two-community systems, we identify a special coupling regime where interaction energies cancel, yielding purely entropy-driven dynamics – a statistical physics representation of atomized societies without structured influence. Monte Carlo simulations confirm these phases and reveal metastable switching dynamics in finite systems. Furthermore, we establish an exact correspondence between this social model and mean-field SU(N) quantum spin systems with quadratic and cubic Casimir interactions, revealing a social-quantum duality. This duality enables quantitative classification of social structures via Young diagrams and reinterprets quantum symmetry breaking as opinion stratification, bridging statistical sociology and quantum many-body physics.

In this study, we explore the fundamental principles behind the architecture of the human brain's structural connectome, from the perspective of spectral analysis of Laplacian and adjacency matrices. Building on the idea that the brain strikes a balance between efficient information processing and minimizing wiring costs, we aim to understand the impact of the metric properties of the connectome and how they relate to the existence of an inherent scale. We demonstrate that a simple generative model, combining nonlinear preferential attachment with an exponential penalty for spatial distance between nodes, can effectively reproduce several key characteristics of the human connectome, including spectral density, edge length distribution, eigenmode localization and local clustering properties. We also delve into the finer spectral properties of the human structural connectomes by evaluating the inverse participation ratios (IPR_q) across various parts of the spectrum. Our analysis reveals that the level statistics in the soft cluster region of the Laplacian spectrum deviate from a purely Poisson distribution due to interactions between clusters. Additionally, we identified scar-like localized modes with large IPR values in the continuum spectrum. We identify multiple fractal eigenmodes distributed across different parts of the spectrum, evaluate their fractal dimensions and find a power-law relationship in the return probability, which is a hallmark of critical behavior. We discuss the conjectures that a brain operates in the Griffiths or multifractal phases.

We consider the orientational diffusion controlled by the hyperspherical Laplacian, ∇^2_D, on the surface of the D–dimensional hypersphere in the limit D →∞. We find that for stretched paths with lengths relatively short compared to the hypersphere's radius, the finite-size corrections in orientational correlations are controlled by the Kardar-Parisi-Zhang (KPZ) scaling exponent, γ = 1/3. In addition, we speculate about the topology of the orientational target space representing the surface of the hypersphere.

Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp--plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling at the critical Euclidean time $t_{E}^*=c_{cr}K$, for the Krylov chain of finite length $K$, and $c_{cr}=O(1)$. In particular, we find a scaling $\sim K^{1/3}$ for fluctuations in the one-point correlation function and a dynamical scaling $\sim K^{-2/3}$ associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is some outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise $O(\frac{1}{N})$ term is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.

The flows generated by the iterative dynamics of triangle reflections are analyzed. These flows are interpreted as the adiabatic dynamics of probe particles within the fundamental domain of the modular group. Two specific cases of lattices are considered: (a) those generated by reflections of equilateral triangles, and (b) those generated by reflections of rectangular isosceles triangles. We demonstrate that the stationary points of the flows for equilateral and isosceles triangles correspond to the “Golden” and the “Silver” ratios, respectively.

Анализируются потоки, порожденные итеративной динамикой отражений треугольников. Эти потоки интерпретируются как адиабатическая динамика пробной частицы в фундаментальной области модулярной группы. Рассматривается два конкретных случая решеток: полученные отражениями равносторонних треугольников и полученные отражениями прямоугольных равнобедренных треугольников. Показано, что стационарные точки потоков для равносторонних и равнобедренных треугольников соответственно являются золотым и серебряным сечениями.

We discuss a two-parameter renormalization group (RG) flow when parameters are organized in a single complex variable, τ, with modular properties. Throughout the work we consider a special limit when the imaginary part of τ characterizing the disorder strength tends to zero. We argue that generalized Riemann-Thomae (gRT) function and the corresponding generalized Devil's staircase emerge naturally in a variety of physical models providing a universal behavior. In 1D we study the Anderson-like probe hopping in a weakly disordered lattice, recognize the origin of the gRT function in the spectral density of the probe and formulate specific RG procedure which gets mapped onto the discrete flow in the fundamental domain of the modular group SL(2,Z). In 2D we consider the generalization of the phyllotaxis crystal model proposed by L. Levitov and suggest the explicit form of the effective potential for the probe particle propagating in the symmetric and asymmetric 2D lattice of defects. Analyzing the structure of RG flow equations in the vicinity of saddle points we claim emergence of BKT-like transitions at Imτ→0. We show that the RG-like dynamics in the fundamental domain of SL(2,Z) for asymmetric lattices asymptotically approaches the “Silver ratio”. For a Hubbard model of particles on a ring interacting via long-ranged potentials we investigate the dependence of the ground state energy on the potential and demonstrate by combining numerical and analytical tools the emergence of the generalized Devil's staircase. Also we conjecture a bridge between a Hubbard model and a phyllotaxis.

We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$, in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices, $A_N$. The most attention is paid to the distribution and typical fluctuations of $\lambda_{\rm max}$ in the vicinity of the percolation threshold, $p_c=\frac{1}{N}$. The overwhelming majority of subgraphs representing $A_N$ near $p_c$ are exponentially distributed linear subchains, for which the statistics of the normalized largest eigenvalue can be analytically connected with the Gumbel distribution. For the ensemble of {\rm all} subgraphs near $p_c$ we suggest that under an appropriate modification of the normalization constant the Gumbel distribution provides a reasonably good approximation. Using numerical simulations we demonstrate that the proposed transformation of $\lambda_{\rm max}$ is indeed Gumbel-distributed and the leading finite-size corrections in the vicinity of $p_c$ scale with $N$ as $\sim \ln^{-2}N$. All together, our results reveal a previously unknown universality in eigenvalue statistics of sparse matrices close to the percolation threshold.

We discuss the generalization of a classical problem involving an $N$-step ideal polymer adsorption at a sticky boundary (potential well of depth $U$). It is known that as $N$ approaches infinity, the path undergoes a 2nd-order localization transition at a certain value of $U_{\text{tr}}$. By considering the random walk on a half-line with a sticky boundary (Model I), we demonstrate that the order of the phase transition can be altered by adjusting the scaling of the first return probability to the boundary. Additionally, we present a model of a random path on a discrete 1D lattice with non-uniform local hopping amplitudes and a potential well at the boundary (Model II). We illustrate that one can tailor such amplitudes so that the polymer undergoes a 3rd-order phase transition.

We discuss the eigenvalue detachment transition in terms of scaling of fluctuations in ensembles of paths located near convex boundaries of various physical nature. We consider numerically the BBP-like (Baik-Ben Arous-P\'ech\'e) transition from the Gaussian to the Tracy-Widom scaling of fluctuations in several statistical systems for both canonical and microcanonical ensembles and identify the corresponding control parameter in each case. In particular, for fixed path length (microcanonical) ensemble of paths located in the vicinity of a partially permeable semicircle, the transition occurs at the critical value of a permeability. The Tracy-Widom regime and the BBP-like transition for fluctuations are discussed in terms of the Jakiw-Teitelbom (JT) gravity with a radial cutoff which, in turn, has an interpretation as a ensemble of fixed length world-line geometrically constrained trajectories of a charged particle in an effective transversal magnetic field.

We consider the microcanonical ensemble of random paths of fixed length, N, in the vicinity of a semi-permeable disc boundary. Varying the penetration probability, η, through the boundary from weakly penetrating (0 < η ≪ 1) to strongly penetrating (η → 1) regimes, we observe the “Baik–Ben Arous–Péché” (BBP) transition which manifests itself in the change of paths fluctuations in the direction perpendicular to the boundary. The span of fluctuations, Δ behaves as Δ(N) ~ N1/3 for 0 < η ≪ 1 and as Δ(N) ~ N1/2 for η → 1. We apply the method of the dimensionality reduction to analyze the statistics of ensemble of trajectories and identify the BBP transition with specific clustering pattern on a two-dimensional plane.

While stretching of a polymer along a flat surface is hardly different from the classical Pincus problem of pulling chain ends in free space, the role of curved geometry in conformational statistics of the stretched chain is an exciting open question. We use scaling analysis and computer simulations to examine stretching of a fractal polymer chain around a disc in 2D (or a cylinder in 3D) of radius R. We reveal that the typical excursions of the polymer away from the surface and curvature-induced correlation length scale as Δ∼R^{β} and S^{*}∼R^{1/z}, respectively, with the Kardar-Parisi-Zhang (KPZ) growth β=1/3 and dynamic exponents z=3/2. Although probability distribution of excursions does not belong to KPZ universality class, the KPZ scaling is independent of the fractal dimension of the polymer and, thus, is universal across classical polymer models, e.g., SAW, randomly branching polymers, crumpled unknotted rings. Additionally, our Letter establishes a mapping between stretched polymers in curved geometry and the Balagurov-Vaks model of random walks among traps.

We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

We demonstrate here a series of exact mappings between particular cases of four statistical physics models: equilibrium 1-dimensional lattice gas with nearestneighbor repulsion, (1 + 1)-dimensional combinatorial heap of pieces, random walks on half-plane and totally asymmetric simple exclusion process (TASEP) in one dimension (1D). In particular, we show that generating function of a steady state of one-dimensional TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of (i) a representation of the steady-state TASEP configurations in terms of (1+1)-dimensional heaps of pieces and (ii) a theorem connecting the partition function of (1 + 1)-dimensional heaps of pieces with that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.