Papers

The current debate on higher-order interactions raises a fundamental question: when can systems organized through group interactions be faithfully represented within a graph-based formalism? Recent results show that graph descriptions can reproduce higher-order dynamics exactly or preserve selected macroscopic observables. Yet reducibility does not necessarily imply simplification, as information removed from the interaction structure may reappear as complexity in the effective dynamics, while structural features such as nestedness, heterogeneity, and cross-order correlations may be obscured by the reduction. We argue that formal representability alone is insufficient as a criterion for model choice. Both "simpler" and "equivalent" are question-dependent notions: parsimony must be assessed for the complete structure-dynamics model, and adequacy must be defined relative to the observables, scales, and regimes required by the scientific question. A reduced model may therefore be fully adequate for locating a phase transition or identifying a universality class while being inadequate for reproducing microscopic trajectories, transient dynamics, or structure-function relations. When group structure carries relevant microscopic information, higher-order representations may remain the most direct, interpretable, and parsimonious description, as well as the natural framework for understanding how microscopic organization gives rise to macroscopic behavior and functionality.

We develop an encounter-based formulation of restricted diffusion in a bounded domain with multiple targets, resolving separately the boundary local time accumulated on each target. Unlike the single-target problem, the target projection operators do not generally commute with the governing Dirichlet-to-Neumann (DtN) operator, so that its eigenbasis does not diagonalize the local-time dependence. We overcome this difficulty by decomposing the DtN operator into target-restricted blocks. Multi-dimensional Laplace inversion then yields a convergent switching expansion, whose successive terms describe alternating diffusive transfers between the targets. We also derive an equivalent operator-valued renewal equation. For two targets, diagonalizing the target-restricted blocks provides an explicit spectral representation in terms of two DtN block spectra and an inter-target coupling matrix. When the coupling preserves spectral modes, the switching series can be resummed exactly in terms of modified Bessel functions. In general, as the off-diagonal DtN block is smoothing for separated targets, one can resort to low-rank finite-dimensional approximations. We examine an effective two-mode reduction for small, well-separated targets and identify a regime in which repeated inter-target transfers are progressively suppressed. Probabilistic interpretations and implications for diffusion-controlled reactions are discussed.

Nucleation rates are exponentially sensitive to the thermodynamic driving force and can therefore depend strongly on the treatment of long-range intermolecular interactions. Here, using the Lennard–Jones (LJ) system as a benchmark, we combine molecular dynamics (MD) simulations, jumpy forward-flux sampling (jFFS), and free-energy calculations to quantify the effect of potential truncation on melting thermodynamics, homogeneous crystal nucleation kinetics, and computational cost. Within the cutoff-radius range $2.5\sigma\le r_c\le 6\sigma$, the melting temperature at zero pressure varies by approximately 11%, while the nucleation rate changes by approximately ten orders of magnitude. By invoking classical nucleation theory (CNT), we show that this pronounced kinetic sensitivity originates primarily from cutoff-induced changes in the chemical potential difference between the liquid and crystalline phases. Building on this observation, we develop a CNT-based framework for extrapolating finite-cutoff rates to the full-potential limit and for estimating the expected rate deviations at other cutoff radii and temperatures. These findings also provide a systematic basis for cutoff selection: the optimal cutoff should minimize computational cost while keeping the deviation from the full-potential rate within acceptable bounds. At $kT/\epsilon=0.5$, $r_c=4\sigma$ provides a reasonable compromise according to these criteria. We further demonstrate that conventional homogeneous tail corrections do not offer a reliable alternative, as they cannot consistently account for the liquid, crystalline, and interfacial environments present during nucleation. Our findings highlight the need to specify and validate the truncation scheme as an integral component of force-field development in simulations of nucleation and other interfacial phase transitions within inhomogeneous environments.

Symmetry plays a fundamental role in modern physics by guiding the classification of phases and the construction of effective theories of critical systems. Identifying collective symmetry remains challenging because tests based on preselected order parameters can overlook hidden components. A central goal is therefore to extract symmetry and critical information from a single accessible component. We pursue this goal using Lee–Yang zero ratios generated by a single ordering field. Unmeasured components shape the projected fluctuation distribution and hence the relative zero positions. Deep in a phase with spontaneous symmetry breaking, we show analytically that the zero ratios for an isotropic order parameter approach a parameter-free Bessel sequence fixed by the number of collective components. Near quantum criticality, fluctuations deform this distribution. The resulting ratio deviations and their finite-size coupling dependence carry additional critical information. We demonstrate these results with large-scale quantum Monte Carlo simulations of many-body systems with conventional and enlarged continuous symmetries. This approach provides quantitative access to collective symmetry and criticality without measuring every competing order.

Universality in nonlinear nonequilibrium systems is typically expressed through scaling laws that render macroscopic behavior insensitive to microscopic details. Whether this universality survives when periodic forcing and nonlocal dissipative memory act simultaneously remains an open question in driven open dynamics. Here, we demonstrate that barrier-crossing processes in periodically driven dissipative systems are governed not merely by modified exponential scaling, but by an explicit two-parameter universal response function. Within a semiclassical instanton framework incorporating Floquet modulation and Ohmic environmental coupling, the tunneling exponent factorizes into a system-dependent static contribution and a universal function of two dimensionless control parameters: normalized driving frequency and dissipation strength. This factorization arises from the combined modification of a single saddle-point trajectory and introduces no additional independent scaling variables. Weak-to-moderate dissipation acts as a smooth dynamical renormalization of the effective action, preserving the saddle-point structure and enabling controlled analytical expansion. In the high-frequency regime, the response exhibits universal dynamical averaging, while an explicit integral representation establishes a continuous adiabatic to Floquet crossover. Direct numerical evaluation of the nonlocal instanton action confirms that the normalized tunneling exponent exhibits a universal dependence on the driving parameters across different model systems. These results identify driven dissipative barrier crossing as a distinct two-parameter universality class within nonlinear nonequilibrium dynamics and provide a predictive functional framework for response phenomena in driven systems with memory.

It is often assumed that general stationary solutions to the wave kinetic equation are described solely by conserved fluxes and temperature. The families of solutions parameterized by these quantities are referred to as warm cascades, and are derivable from differential models of wave kinetics. These models rely on the assumption of local interactions, and we demonstrate by a time scale argument that the nonlocal effect of thermal damping on the cascade may result in dependence on an additional parameter, the UV cutoff, calling into question the validity of warm cascade solutions. This is shown for systems with both three-wave and four-wave interactions. We then consider the depletion of the cascade by this damping, and present a numerical demonstration of this nonlocal and nonuniversal behavior.

Conformal interfaces provide an important setting for studying universal transmission phenomena in one-dimensional quantum critical systems. While energy and information transmission across such interfaces are characterized by universal quantities in closed systems, the corresponding role of conformal interfaces in dissipative dynamics is less understood. In this work, we study relaxation in locally dissipative quantum critical chains with a conformal interface and show that a universal characterization emerges at the level of individual relaxation modes. We first show that the relaxation coefficient defined in the recent work [1] from the Liouvillian gap can become non-universal for certain boundary conditions because the mode determining the smallest decay rate can change as the interface transmission is varied. To resolve this ambiguity, we introduce a mode-resolved relaxation coefficient $c_{\rm relax}$ by continuously tracking the same Liouvillian rapidity mode as a function of the interface transmission. Using analytical and numerical calculations for a critical harmonic chain and a critical free-fermion chain, we find that, in the weak-dissipation regime, $c_{\rm relax}$ follows the same universal dependence on the interface transmission in all cases considered, independent of microscopic details such as the boundary conditions, dissipation strength, and location of the local dissipation. For boundary dissipation, this universal behavior persists even at finite dissipation strength. Our results establish a universal mode-resolved characterization of relaxation across conformal interfaces in open quantum critical systems.

Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.

Counting the number of coverings of $s$ dimers on two-dimensional quadratic lattices is considered as intractable and belongs to \#P-complete class. We reveal the structure of the exact solution to the problem and provide an explicit formula for it, which includes $s-1$ nesting sums. This results in an exponential time complexity of $O(2^s)$. The solution is explicitly determined by a sequence that exhibits double-exponential growth.

Emergent coordinated behaviors of AI agents are starting to present critical safety risks. A key phenomenon driving these behaviors is the rapid formation and spread of beliefs about the world, and mechanistic understanding is crucial for collective alignment. To this end, we introduce the Flag Game, a toy model for studying the mechanisms of collective belief formation. Concretely, a hidden country flag defines the ground truth, and each bounded agent directly observes only a private crop but can exchange beliefs and weigh social evidence from peers. Despite its simplicity, the Flag Game reproduces rich collective phenomenology: non-monotonic scaling of performance with population size, accuracy gains from social-awareness prompting and team diversity, and strong effects of organizational structure. In particular, we identify that collective belief collapse at small population sizes turns into collective belief polarization as the population grows. This polarization causes the performance decline at large population sizes, but creates diversity in collective beliefs. Finally, we dissect the mechanisms underlying collective belief collapse and polarization with two complementary approaches. We first introduce social circuit attribution, a technique to predict which agent, and what view, matters most to collective dynamics, and verify its predictions by causal interventions on agents, tracing how agent patching changes collective outcomes. However, the efficacy of causal interventions on agents decreases as the population grows. We therefore develop a statistical mechanical theory for larger populations and verify that it matches the empirical phase diagram. Together, these results take a first step toward mechanistic swarm interpretability, a science of how the properties of individual agents and their communication give rise to emergent collective behavior.

We revisit the hierarchy of relaxation time scales in stochastic kinetically constrained models (KCMs) obeying detailed balance, using an expansion method developed recently for their quantum counterparts. In the classical setting, we expand the stochastic generator in the low-temperature equilibrium concentration of excitations. As in quantum KCMs, successive truncations of the expansion reveal a nested hierarchy of metastable configurations that remain frozen on progressively longer time scales. Applying the method to the high-to-low temperature quench in the classical one-dimensional East and Fredrickson-Andersen models, we recover the known hierarchy of metastable plateaus and the associated perturbative time scales. We find that the hierarchical time scales are related to the smallest domain lengths. Our results show that the method developed for quantum kinetically constrained models can be used to provide a systematic description of slow relaxation in their classical counterparts as well.

For a diffusive system on a ring, with a battery at the origin, we use the macroscopic fluctuation theory (MFT) to obtain the large deviation function of the current. In the case of non-interacting particles, the predictions of the MFT agree with the result of a direct exact microscopic calculation. To our surprise, the large deviation function of non-interacting particles with a battery has the same form as for the Symmetric Simple Exclusion Process (SSEP) with open boundary conditions. For the SSEP on a ring with a battery, the predictions of the MFT seem to agree with the results of long Monte Carlo simulations, although much longer simulations would be needed to better test the validity of these predictions.

In this introductory review, we present and compare four algorithms for the numerical integration of the Lindblad equation for one-dimensional quantum lattice systems. All four methods are based on Matrix Product State representations and can be viewed as extensions of the Time-Evolving Block Decimation (TEBD) algorithm to open quantum systems. Two approaches directly integrate the vectorized Lindblad equation, one of them explicitly enforcing the positivity of the density matrix. The other two rely on stochastic unravelings of the Lindblad equation, namely the quantum trajectory and quantum state diffusion approaches. We discuss the principles, numerical implementation, accuracy, and computational efficiency of the different methods, and benchmark them against an exactly solvable free fermion model.

In this paper, we study the effect of an external random field on the localisation / delocalisation of the long-range Discrete Gaussian Chain (DGC), recently studied by Garban arXiv:2312.04536 and Coquille et al. arXiv:2412.15782. We prove that the model is delocalised at every inverse temperature for any polynomial decay $\alpha > 3/2$. This behaviour differs qualitatively from the one of the disorder-free long-range DGC, for which delocalisation at every temperature occurs only for $\alpha > 2$, similarly to what happens for long-range Ising models as already proved by Aizenman and Wehr. Additionally, we provide a more detailed description of the delocalised phase by identifying the growth exponent and scaling limit of the height at the origin for any decay exponent $\alpha > 3/2$.

Long-range (LR) interactions are no longer just a theoretical idea: they can be engineered in several low-dimensional platforms and offer a new way to control heat transport. At the same time, they push beyond the standard picture developed mainly for short-range, momentum-conserving lattices. In this review, we consider one-dimensional Fermi-Pasta-Ulam-Tsingou (FPUT)-type lattices where LR effects enter mainly through a quartic anharmonic coupling that decays as a power law, characterized by an exponent {\sigma}. Our goal is to explain, in a unified way, how LR anharmonicity changes microscopic energy exchange, collective dynamics, and the macroscopic scaling laws of heat conduction-while also clarifying what is well established and what is still debated (For more details of the abstract, please see the main text and paper).

Over the past decade, high-order series expansions based on linked-cluster methods have become an important tool for studying low-energy properties of gapped quantum systems with long-range interactions. We introduce a deterministic framework that removes a central computational bottleneck of this method. Our graph zeta method replaces the costly and statistically noisy Monte Carlo (MC) evaluation of high-dimensional lattice sums by a systematic, high-precision computation that delivers series coefficients within minutes on standard desktop hardware. The full momentum-dependent series is obtained in a single calculation, enabling high-resolution excitation spectra throughout the Brillouin zone. Building on the companion paper [1], the method reformulates graph-embedding sums as graph zeta functions and decomposes them into blocks classified by their treewidth tw. Low-treewidth blocks (tw$\leq2$) admit closed expressions based on Epstein zeta functions, while higher-treewidth blocks (tw$>2$) are evaluated using tensor-network bucket elimination. We benchmark the approach for transverse-field Ising models with power-law interactions in 1d, 2d, and 3d, reproducing previous MC results at a fraction of the computational cost while enabling substantially denser parameter sampling. An open-source implementation makes the method directly applicable to general interactions and large parameter scans. As an application, we compare microscopic interaction models for the stacked quasi-2d transverse-field Ising triangular-lattice antiferromagnet KTmSe$_2$ and find that a model including dipolar interactions best describes existing experimental data. The graph zeta method thus turns high-order linked-cluster expansions into a practical and deterministic tool for fast quantitative momentum-resolved modeling of short- and long-range quantum matter.

Active matter systems can generate mechanical stresses that deform their surroundings, providing a route to transport and shape dynamics far from equilibrium. Deformable vesicles containing active particles offer a minimal setting in which such active stresses are directly coupled to boundary mechanics. Here, we use numerical simulations to investigate a two-dimensional, fixed-area vesicle filled with chiral active particles interacting through excluded volume and polar alignment. We find that the coupling between particle chirality and vesicle deformation produces distinct modes of collective motion, including run-and-tumble-like migration, rotor-like dynamics, and persistent spinning. Most notably, we show that vesicle rotation is non-monotonic in chirality: at fixed activity, an optimal chirality maximizes the rotational velocity. We develop an analytical theory that relates the vesicle rotation to the effective torque generated by the chiral active particles. The theory identifies the competition underlying the optimal chirality and predicts a scale-invariant dependence of the rotational velocity on activity and chirality. This scaling collapses simulation results obtained over different activity strengths onto a universal curve. Our results establish a general mechanism by which chirality and collective alignment regulate the transmission of active stresses to deformable boundaries, providing a framework for understanding transport and rotational dynamics in confined active systems.

Active microrheology probes a crowded medium by forcing a tracer and measuring its response. In single-file transport, where particles cannot overtake, a constant force $F$ produces a subballistic displacement $\langle X_t\rangle\simeq \sqrt{t}\,\xi(F)$ together with a persistent bath deformation. Despite decades of work, the exact nonlinear response at arbitrary force has remained confined to a few special solvable models. Here we remove this restriction by combining recent advances in hydrodynamic transport coefficients, a pressure-balance formulation of the local drive, and single-file duality, which eliminates the resulting moving boundary. This yields a closed exact boundary-value problem for general diffusive single files, determining both $\xi(F)$ and the full density profile. For overdamped Brownian particles with general interactions, all microscopic interaction details enter only through the equilibrium equation of state, bringing realistic interacting systems, including finite-width quasi-one-dimensional channels, within exact reach. The solution also reveals universal global laws; in particular, the bath-density dipole is fixed by the applied force independently of the interaction potential.

We investigate the connection between classical and quantum chaos in a dissipative system of two coupled collective spins. Although the classical dynamics possesses a stable fixed-point attractor, we find pronounced signatures of chaos in the transient dynamics preceding relaxation. In the quantum model, such transient chaos is accompanied by a transition to random-matrix statistics in the relevant part of the Liouvillian spectrum and by a qualitative increase in the number of Liouvillian modes contributing to expectation values of physical observables. The transition occurs before the bifurcation at which the stable fixed point disappears and a chaotic attractor emerges. Our results show that Liouvillian quantum chaos can be associated with transient classical chaos, rather than requiring a chaotic asymptotic attractor, and provide a refinement of the Grobe-Haake-Sommers conjecture.

The survey discusses the problem of measuring correlations in composite quantum and statistical systems. After briefly recalling the standard methods of describing correlations, the emphasis is on the method of correlation indices. The latter method for quantifying the strength of correlations in composite physical systems is sufficiently general allowing for measuring in a unique way the correlation strength in both quantum and statistical systems, for static as well as for dynamic processes. The correlation indices, can be defined for correlation operators, reduced density matrices, and other operators containing information on correlations in the studied composite systems. The correlation indices measure all types of correlations, quantum correlations, such as entanglement, as well as classical correlations. They are applicable for characterizing static as well as dynamical processes. Examples are given of correlation indices for several quantum states and for spin and pseudospin systems. The quantification of correlations in a nonequilibrium system is exemplified by calculating the correlation index for a trapped Bose-Einstein condensate subject to the action of an alternating field.

It has been over fifty years since Kenneth Wilson had his Nobel-prize-winning ideas on the renormalization group. In this time frame, despite many attempts, no mathematical results have implemented Wilson's vision to a satisfactory degree. Although a number of predictions stemming from the renormalization group framework have been proven to date, these proofs usually rely on alternative ideas and do not cover the full predictive power of Wilson's renormalization group. The discovery of pathologies within rigorous implementations of renormalization group transformations have further reduced the enthusiasm of the mathematics community for this subject. This paper presents a novel framework to implement Wilson's original idea in a rigorous manner based on inverse limits of finite systems. Doing so, one avoids these pathologies and hence obtains a mathematically sound framework for Wilson's renormalization group. This opens up a completely new path to tackle some of the existing problems in rigorous renormalization group theory like the existence of non-trivial fixed points.

We study statistical variants of the classical largest average submatrix problem. For small submatrices (where the dimension is less than linearly proportional to the original matrix), the problem is typically well understood and believed to exhibit the statistical-computational gap (SCG). However, analytical treatment of both the information-theoretic and algorithmic aspects of the linear regime remains challenging, and no mathematically rigorous results have yet arrived anywhere close to proving or disproving SCG existence in this setting. Focusing on the linear regime, we make strong progress in several key directions: 1) We develop a generic Random Duality Theory (RDT) framework to characterize largest average submatrix values. 2) Using the plain RDT variant, we obtain closed-form upper bounds as explicit functions of dimensionality proportionality parameters. 3) We demonstrate that a lifted RDT variant strictly improves upon the plain RDT within a certain linear range of submatrix dimensions. 4) For small submatrices where the dimensional proportionality constants approach zero, we prove that our results match both the replica results from [34] (obtained via one-step replica symmetry breaking) and the sublinear results from [18,45].

Can quantum entanglement continue to grow even after the entanglement entropies of subsystems have saturated? We address this question using genuine multi-entropy, which probes multi-partite entanglement structure beyond bipartite entanglement. In a two-sided BTZ black hole, this quantity grows linearly with time after every subsystem entropy of a fixed tripartition has saturated, extending the interval in which entanglement detects Einstein-Rosen bridge growth. The result refines the ER=EPR relation by identifying multi-partite structure invisible to ordinary entanglement entropies. Some spin-chain models and the spatially partitioned Sachdev-Ye-Kitaev (SYK) model exhibit distinct transients, while the late-time plateau values in holography, the SYK model, and Haar-random states share a universal behaviour.

We study the Loschmidt echo in a system of N non-interacting spinless lattice fermions released from a double-domain-wall initial state. In the large-N limit, the return probability is characterized by a large-deviation rate function known as the dynamical free energy. The Loschmidt echo is dominated by a complex instanton configuration, and the dynamical free energy develops a logarithmic singularity at the dynamical quantum phase transition. We obtain analytical results in the short-time and long-time regimes and support them with numerical computations. We show that the transition can be understood as a topological change of the dominant instanton configuration, analogous to the emergence of a cut in random matrix theory. We further show that post-selection can drive the system through two successive DQPTs, associated with successive topological changes of the dominant instanton configuration in complex time.

We present a general method for estimating entropy differences between arbitrary probability distributions using stochastic interpolants. The marginal distributions which bridge the base and target obey a continuity equation, which allows a straightforward calculation of the entropy difference in terms of an inner product of the probability flow velocity and score fields. This formulation has several advantages: (i) no computationally expensive divergence calculations of either field are required, (ii) the score field need not even be learned directly if model transferability is not required, and (iii) on-the-fly estimates are produced nearly for free during training. When tractable base distributions are chosen (e.g. Gaussian chain, ideal gas, etc.), statistical thermodynamic entropies are then immediately recovered. Notably, the analysis relies only on the definition of the Gibbs-Shannon entropy, rather than any particular statistical mechanical ensemble, so that generalized non-equilibrium entropies may be computed. The method is demonstrated on several systems of increasing complexity: (i) a 40-dimensional Gaussian mixture model, (ii) the classical XY model of N spins arranged in one dimension, (iii) the 13-atom Lennard-Jones cluster, and (iv) an active Brownian polymer.

The quantum Mpemba effect is the counterintuitive phenomenon whereby a quantum state initially farther from equilibrium relaxes faster than one initially closer to it. We demonstrate this effect in random Clifford circuits with a conserved $\mathrm{U}(1)$ charge, using the entanglement asymmetry to characterize relaxation through dynamical symmetry restoration. Exact numerical simulations show that an initially more asymmetric state can become locally more symmetric than an initially less asymmetric one. We explain this behavior by mapping the dynamics onto a charge-conserving quantum automaton, in which the decay of the entanglement asymmetry is controlled by the statistics of encounters between two particle species evolving according to a symmetric simple exclusion process. This mapping yields an analytic expression for the entanglement asymmetry and provides a simple microscopic mechanism for the quantum Mpemba effect: stronger initial symmetry breaking corresponds to a denser particle configuration, which enhances the frequency of particle encounters and thereby accelerates symmetry restoration.

Can the terminal macroscopic outcome of a many-body system be inferred from its microscopic initial data without simulating the full trajectory? Rather than construct such a shortcut, we address a more fundamental question for Gaussian microscopic inputs: can any fixed Wiener-Hermite degree retain a nonvanishing fraction of the variance of the terminal outcome as the system grows? We consider homogeneous systems with independent Gaussian disorder in which all microscopic coordinates are symmetry-equivalent, the terminal event is monotone in each disorder variable, and a uniform disorder shift is exactly equivalent to a control-field shift with a size-independent conversion factor. Using forward and inverse Gaussian influence bounds together with a Gaussian Russo formula, we derive a directly measurable criterion that is both necessary and sufficient for noise sensitivity. Specifically, the correlation between the original and coordinate-perturbed terminal outcomes vanishes asymptotically for every fixed nonzero level of coordinatewise noise if and only if the slope of the outcome probability with respect to the control field at the balanced threshold grows more slowly than the square root of the system volume. Event-driven simulations of the three-dimensional driven random-field Ising model up to linear size 192 find that both the normalized response and the correlations between perturbed samples decrease overall, consistent with the noise-sensitive regime at finite size. For spatial Stag-Hunt dynamics with prescribed seeds, the criterion generalizes through an effective number of influential coordinates. Separately, simulations of a path-dependent best-response game show, over the sizes studied, that the terminal equilibrium can depend on the update schedule while still carrying substantial finite-order predictive information.

In this paper, we study Renyi and Tsallis information entropies for a Hamiltonian system with position-dependent mass confined in harmonic oscillator potential. Gegenbauer polynomials are used to obtain the position eigenfunction of such a system, and the modified Bessel function of the second kind is used to determine the equivalent momentum eigenfunction. By means of probability densities of both representations, we analytically and numerically evaluate the Heisenberg-like uncertainty of this system. Because the Renyi and Tsallis information entropies in position representation are described by integral functionals of the Gegenbauer polynomials, these quantities are much more difficult to calculate. To get around this problem, we evaluate these information entropies at the system s asymptotical limit, which corresponds to the behavior of an undeformed harmonic oscillator. Nevertheless, no approximation method is used to obtain the Tsallis and Renyi information entropies in momentum representation. In both representations, we find that these information entropies approach the Shannon entropy when the entropic parameter alpha-> 1 and are closed to the results of similar models of the literature. Finally, we evaluate the related entropic uncertainty relations numerically to validate the latter obersevations.

The free-energy principle (FEP) and active inference describe adaptive systems through probabilistic generative models and free-energy functionals, whereas integrated information theory (IIT) characterizes intrinsic cause-effect organization. We separate two questions: whether revised IIT quantities admit local free-energy-like representations and whether scalar expected free energy can determine system integrated information. We show that the 2026 IIT intrinsic-differentiation and specification terms admit exact surprisal representations, and hence variational-free-energy representations under exact inference. We then construct noisy binary permutation networks whose causal topology varies while local transition reliability is fixed. In a homogeneous state, all permutations yield the same one-step predictive distribution and state-risk expected free energy under common preferences and observation models, yet revised IIT distinguishes them: multicycle permutations have zero whole-system integration, whereas a single noisy cycle has positive integration at intermediate reliability. For an n-unit noisy single cycle, we derive a closed-form expression for revised system integrated information, prove that the minimum-information partition cuts exactly two causal edges, and show that as n grows the optimal reliability approaches unity, the joint specified-state probability approaches one half, and the maximum integration approaches one ibit. Numerical IIT calculations for n = 2-5 and exhaustive partition enumeration through n = 8 reproduce the analytical results. Thus, local surprisal/free-energy correspondence coexists with global non-equivalence: equal expected free energy does not imply equal revised IIT system integrated information. Any synthesis of IIT with FEP/active inference must retain counterfactual causal information beyond a scalar free-energy value.

For the neutral periodic one-dimensional Riesz gas with pair potential locally $-|x|^a$, $0<a<1$, we prove a particle-displacement variance bound of order $\beta^{-1}$, uniformly in the number of particles. Log-concavity also gives exponential displacement tails. Every stationary periodic thermodynamic limit is simple, has intensity one, and admits a stationary ordered matching to the unit lattice with the same bounds. Each limit satisfies the canonical Gibbs equations for the full-line interaction, with an ordinary symmetric spatial principal value for the exterior potential. The matching implies uniformly bounded interval number variance and a positive limiting second moment of the reciprocal-lattice Fourier average at sufficiently low temperature. The main estimate compares the inverse random Hessian, through deterministic electrical flows, to a transient long-range network.