Papers

A universal characterization of non-equilibrium steady states in interacting quantum many-body systems remains one of the central challenges of statistical physics. Here, we address this problem for a paradigmatic model of diffusive interacting quantum matter—the boundary-driven XXZ spin chain with bulk dephasing—and derive, directly from its microscopic Lindblad dynamics, an emergent classical Macroscopic Fluctuation Theory (MFT) governing its large-scale fluctuations. Crucially, the resulting hydrodynamics carries a density-dependent diffusivity and mobility as the fingerprint of interactions. This effective description enables the exact computation of the stationary density profile, long-range correlations, and the full counting statistics of the current, in excellent agreement with tensor-network simulations. Our work demonstrates that noisy quantum many-body systems can realize the universality class of genuinely interacting diffusive matter, beyond the constant-diffusivity class of the symmetric simple exclusion process, and establishes MFT as a powerful universal framework for interacting diffusive quantum systems.

Non-commutative sphere regularization of CFT has become a powerful tool for computing conformal data of 3d CFTs and was recently extended to 4d CFTs. In this work we further construct 5d free-scalar CFT and Yang-Lee CFT on a fuzzy 4-sphere. We observe a continuous phase transition and conformal towers in these examples. We obtain the first three scalar primaries of the 5d free-scalar CFT, furthermore, we compute the 5d Yang-Lee $\Delta_\phi,\Delta_{\phi^3}$ with $N_e=10$ fermions using exact diagonalization (ED) and $N_e=20$ fermions using density matrix renormalization group (DMRG), obtaining relatively good agreement with both $\epsilon$-expansion and the Gliozzi truncated bootstrap result.

The kinetic Wulff construction predicts symmetric nonequilibrium shapes when crystallographically equivalent facets share a fixed growth rate. However, nanocrystals grow through finite facets whose nucleation barriers and ligand coverages depend on facet size. Here we develop a size-dependent kinetic construction and show that as a consequence infinitesimal seed asymmetries can be amplified into strongly asymmetric nanocrystal morphologies even when all symmetry-related facets obey the same microscopic growth law. Spatially heterogeneous nucleation rates and boundary limited growth generate facet growth velocities that depend on size, and deterministic shape evolution translates these local rates into global shape symmetry breaking. We illustrate the mechanism of persistent anisotropic growth on square and triangular lattices in two dimensions and on FCC seeds in three dimensions, where cuboctahedra can evolve toward rods or tetrahedra depending on which facet-area perturbations are amplified.

We study Bayesian tracking of a diffusing target monitored by a noisy distributed sensor array. Building on an earlier mapping to KPZ growth with a moving defect (or an equivalent directed polymer pinning problem) we determine the phase structure, beyond the previously-studied one-dimensional case, for both Bayes-optimal and suboptimal inference. In $d=2$, theoretical analysis and numerical simulations both give a depinning transition between a successful tracking phase and a failure phase. Weak-coupling RG shows that Bayes-optimal tracking is always successful in $d=2$, but failure can arise from overconfident (suboptimal) inference. In $d=3$, tracking can succeed, or can fail in two distinct ways: the posterior probability distribution may delocalize (no detection), or may become sharply localized, but at the wrong position (a false detection). The two possibilities correspond to Edwards-Wilkinson or Kardar-Parisi-Zhang statistics for the log-posterior. The three phases meet at a Nishimori-like multicritical point on a Bayes-optimal line in a two-parameter phase diagram. (Model misspecification alone can drive depinning into either unpinned phase: underconfidence gives diffuse failure, while overconfidence gives localized-but-wrong failure.) We analyze the transitions between the various phases numerically and with renormalization group arguments. We show that some of these have unusual critical behavior, which the conventional $\epsilon$ expansion fails to describe. Recent rigorous results for directed polymers indicate an alternative scenario. Many of our results, including a scaling relation for exponents at pinning transitions and results for RG flows, are relevant to other phase transitions that involve surface growth or directed polymers in 2+1D or 3+1D.

Kinetically constrained systems, such as those representing Rydberg atom arrays in the blockade regime, have gathered considerable attention due to the presence of atypical eigenstates in their spectrum. The latter manifests itself through the presence of quantum many-body scars as well as unusually large zero-mode (ZM) subspaces which contain analytically tractable eigenstates with various entanglement scalings. In particular, some of the latter are excitable zero-modes (EZMs), meaning that they can be promoted to a non-zero energy for open boundary conditions. In this work, I present an efficient protocol for finding analytical expressions for translation-invariant EZMs in constrained systems, based on the eigendecomposition of the local unconstrained Hamiltonian. I demonstrate the power of this method on a decorated Rydberg chain. In that model, my protocol directly produces a continuous matrix-product-state manifold located entirely in the zero-energy eigenspace for periodic boundary conditions. The span of that manifold grows exponentially with the system, and for odd system sizes it covers the entire zero-momentum eigenspace with zero energy. I then show how the physical structure of the manifold, which is tied to the local eigenbasis, also allows one to derive analytical expressions for ZMs at non-zero momentum and for a polynomial number of exact scars at $E=\pm\sqrt{3}$.

We study the free energy of a critical vector model at large $N$ on $S^{1}\times S^{2}$ with an angular velocity $\hat\mu$ without the singlet constraint. We study the model for which the large $N$ dynamics is controlled by the uniform saddle point of the auxiliary field arising in the Hubbard-Stratanovich transformation. The leading high-temperature behaviour is determined analytically both as an expansion about $\hat\mu r=0$ and $\hat\mu^{2}r^{2}=1$ where $r$ is the radius of the sphere. We supplement the analytic results with a numerical analysis that agrees with both the analytical expansions in their respective regimes and smoothly interpolates between them. The leading high-temperature contribution to the free energy develops a pole at $\hat\mu^{2}r^{2}=1$, in agreement with expectations from the thermal effective field theory. Its residue coincides with that of the massless free theory. Sub-leading terms, however, exhibit non-analytic dependence on the angular velocity and distinguish the critical fixed-point result from the free theory answer. The residue at the pole can also be obtained by placing the model on the pp-wave geometry. We show that the residue agrees with that obtained from the direct computation. The free energy of the model connects the non-trivial fixed point of the $O(N)$ model at $\hat\mu r=0$ to its free fixed point at $\hat\mu^2r^2=1$.

In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.

Cluster algorithms, such as the Swendsen–Wang and Wolff methods, are among the most successful MCMC methods for mitigating critical slowing down in statistical systems. These constructive cluster algorithms, however, fail in the presence of even extremely weak frustration. Here, we sidestep this fundamental limitation by learning rather than constructing the relevant clusters. Specifically, we use the wavelet conditional renormalization group (WCRG) sampling method to learn the probability distribution of collective fluctuations of a frustrated two-dimensional soft-spin model. Configurations are then generated recursively from coarse to fine scales by sampling conditional wavelet distributions. The WCRG method reproduces the main statistical properties of the system across different phases, including the local-field distribution and the structure factor. At an Ising-like critical point, the conditional dynamics remains decorrelated within $\mathcal{O}(1)$ sweeps at each scale, yielding an overall sampling complexity of $\mathcal{O}(\log_2 L)$, thus making WCRG much more efficient than standard local MCMC methods. These results show that learned multiscale sampling can overcome critical slowing down in frustrated systems for which conventional cluster algorithms fail. By assessing the sampling accuracy of different observables, we also clarify the main tradeoff of the WCRG method: the accuracy of the fast sampling scheme depends on the expressiveness of the energy-based model used to estimate the wavelet conditional distributions.

Thermodynamics is operational in the sense that the very concept of thermal equilibrium depends crucially on the choice of observables, while the amount of extractable work is defined relative to the allowed operations. Here we show that isolated quantum many-body states admit a thermodynamic hierarchical structure of observables and allowed operations, where the same state can be thermal at one level of the hierarchy and athermal at another. Enlarging the set of allowed operations therefore renders such hidden athermality a potential resource for work extraction; the resulting work gain is bounded in terms of the difference between the entropy densities of the two levels. This entropy difference takes the form of the mutual-information density. We establish the bounds for three extensions of operational access: increased spatial resolution, increased duration of control, and nonlocal connectivity, which provide access to position-state Holevo information, correlations between neighboring regions and spatially nonlocal correlations, respectively. Thus the difference between entropies at different levels of the thermodynamic hierarchy governs the bound on the work gain associated with moving to a less restricted level.

The trilayered (A-B-A type) anisotropic (single site) XY magnetic model is considered with the intra-plane ferromagnetic and the inter-plane antiferromagnetic interactions. The equilibrium properties of such trilayered anisotropic XY system has been studied by Monte Carlo simulation with Metropolis single spin (randomly chosen) update algorithm. In a given range of interaction parameters, the system has been found to exhibit the high temperature critical (vanishing sublattice magnetisations and total magnetisation) and low temperature compensation (vanishing total magnetisation even for non-vanishing sublattice magnetisations) behaviours. The compensation temperature and the critical temperature both are found to increase with the increase of single site anisotropy. The comprehensive phase diagram is drawn. The finite size study reveals the growth of correlations near the critical temperature.

We reformulate the relaxation problem in statistical mechanics by making explicit what are the operational objects subject to relaxation: the local time statistics of the recorded signal $Z(t)$. These local time statistics are simply the estimated histograms of observations $\{Z(t_i)\}_{i=1}^M$ performed at uniformly random times $\{t_i\}_{i=1}^M$ by a clockless observer. The subject of prediction is a belief about a future fresh out-of-sample reading of a measurement outcome whose distribution is inferred from the mathematical model believed to be true. For finite bounded systems of $N\ge 1$ degrees of freedom global irreversible relaxation of predictions can occur but special initial conditions exist. The form of the predictions depends on certain loss functions whose choice is up to the particular observer. Finally, entropy is given a learning interpretation as mutual information between the observer and the unknown past of the system under consideration and, in complete generality, its stationary value depends on the information available.

Diffusion processes involving membranes are of fundamental importance in both science and technology, since membranes serve as selective interfaces that regulate the transport of mass, charge, and information across multiple length and time scales. In this study, we employ molecular dynamics (MD) simulations to calculate the self-diffusion coefficient, tetrahedral order parameter and hydrogen-bond (HB) lifetime of SPC/E water over a wide high-temperature range in the presence of thermally activated membranes (TAMs). These membranes exhibit thermally controlled stochastic behavior that locally influences particle dynamics by probabilistically inducing elastic scattering events as particles traverse the membrane. The stochastic behavior of the membranes is governed by a sigmoidal profile, which depend on the reduced temperature of the system. The effective activation energy for diffusion is estimated for several membrane configurations. It is found that the diffusion coefficients generally decrease with an increase in the number of membranes and are in reasonable agreement with the Arrhenius approximation at high temperatures. Additionally, both the tetrahedral order parameter and the HB lifetime are only locally sensitive to the action of the membranes.

We investigate information-flow generation in a nonequilibrium two-bead system coupled to two heat baths. We show that the system acts as an information-flow generator in both overdamped and underdamped regimes, with the underdamped dynamics revealing a divergence of the scaled information flow along specific paths in the thermal asymmetry–inertia parameter space that is hidden in the overdamped limit. This information generation suggests a possible route toward information engines and demon-like mechanisms in nanomachines. A symmetry-perturbation analysis of the response landscape of information flow reveals a geometric structure reminiscent of a Ginzburg–Landau framework: the symmetric reference state can correspond to a minimum or maximum depending on the perturbation direction, while the flat overdamped landscape develops a finite curvature under inertia. Mass asymmetry shifts the resulting maxima, and Hessian eigenvalue and eigenvector analysis reveals level touching of principal modes and bimodality along a constant-diffusion path. These results establish a minimal framework for understanding how inertia and microscopic heterogeneity shape information landscapes in nonequilibrium systems, with potential extensions to more complex heterogeneous networks.

We investigate a nonequilibrium steady-state model consisting of two coupled beads with arbitrary masses in contact with two thermal baths at different temperatures. Using a covariance-matrix approach together with numerical simulations of the underdamped Langevin dynamics, we characterize steady-state probability distributions, heat transport, and entropy production. We show that irreversibility measures such as entropy production and heat current are invariant under an exchange of the bead masses, whereas energy-storage observables depend explicitly on the mass arrangement in a symmetrical set up. This reveals a fundamental distinction: energy observables exhibit path dependence in singular mass limits, while transport and irreversibility remain well defined. We show that kinetic temperatures provide the natural variables governing the thermodynamics of the system: their difference controls transport and entropy production, while their sum determines the mean energy via a model specific generalized equipartition relation. In the infinite-mass limit, only constitutive relations expressed in terms of kinetic temperatures remain meaningful. Thus, energy, transport, and irreversibility are unified through kinetic temperatures as the organizing variables. We also derive an effective temperature that defines an equilibrium-like canonical distribution. Finally, we analyze the notion of ergodicity and show that the time-averaged observables converge significantly faster than the ensemble averages.

In this article, we extend learning rules from real binary to complex-valued spins. This formulation allows for a robust and natural representation of grayscale patterns, where spins behave as multi-state neurons and can be stored in a complex-valued weight matrix. We describe a rule that performs better than standard methods, such as Hebbian learning, to encode information in a suitable form for pattern retrieval with networks of oscillators. Since in neural networks it is of interest to have local and incremental learning rules, we prove the extension of a result by Diederich and Opper with our complex-valued spin formulation. We then test the behavior of the associative memory for the system of oscillators under different circumstances for both real-valued, as well as complex-valued correlated and random patterns.

In a slowly varying aperiodic potential $h_i=h\cos(2\pi\beta i^n+\phi)$ with random power-law density interactions $V_{ij}/|i-j|^\alpha$, the resonant-object ensemble changes across $\alpha=2n$, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling $x_{\rm pair}^{4-2n-\alpha}\ln x_{\rm pair}\propto h^2/(Vt_0)$ is unchanged~\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics — the supply $N_0\propto L^{2-n}/h$, the correlated common-phase comb, and the closed-form Hartree variance — proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at $L\le18$ shows that the order-one forward-branching scale $h_{\rm FB}\propto L^{2-n}$ carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged $q\ln(1/q)$ law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold $V_*(\alpha)$ and the boundary-healing recursion; and we treat the marginal case $\alpha=2$, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings — $\alpha=1/2$ (the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate), $\alpha=2n$ (change of the local resonant objects), and $\alpha=2$ (marginality of the long-range shell sum) — none of which is, by itself, a localization transition.

For fermions in a slowly varying aperiodic potential with random interactions $V_{ij}/|i-j|^{\alpha}$, the potential fixes the resonance supply $N_0 \propto L^{2-n}/h$. On crossing ${\alpha}-2n$, wherever Hartree fragmentation is operative, the resonant-object ensemble reconstructs: turning-point runs fragment into clusters whose intrinsic matching law resolves no logarithmic singularity, while isolated wing bonds survive at finite density and restore a reduced logarithm. The scaling $x^{4-2n-{\alpha}}_{pair} ln{x_{pair}} \propto h^2/(V t0)$ is unchanged. Random coefficient fluctuations sustain the $R^{-{\alpha}}$ coupling; uniform ones cancel to $R^{-{\alpha}-2}$.

We introduce a Hamiltonian Active Brownian Particle (HABP) model that connects equilibrium dynamics with the non-equilibrium, two-dimensional overdamped behavior of standard Active Brownian Particles (ABPs). In equilibrium, the system follows overdamped Langevin equations that strictly satisfy the fluctuation-dissipation theorem. Coupling translational and rotational degrees of freedom to separate heat baths ($T_{\theta} > T_{\textrm{tr}}$), drives the system out of equilibrium. In free space, matching the diffusion coefficients yields quantitative agreement with the ABP model in the limit $T_\theta/T_{\textrm{tr}}\to \infty$, whereas in a harmonic potential, this matching is recovered even at finite temperature ratios. Entropy production analysis demonstrates that in the ABP limit, all energy injected by the swim force dissipates into the translational bath, leaving the rotational bath as a zero-cost entropy source. These insights provide the first steps toward equilibrium-inspired descriptions of active matter, facilitating the development of new theoretical frameworks to study activity-induced fluctuations, transport, and phase behavior in living and non-living soft matter systems far from equilibrium.

Reaction-diffusion systems with two species that mutually annihilate through $A+B\to\emptyset$ reactions display a slow decay of the total density $\rho$ which follows an anomalous power law in low dimensions. For hardcore particles in one dimension, this decay follows $\rho \sim t^{-1/4}$ under symmetric diffusion and $\rho \sim t^{-1/3}$ under asymmetric diffusion without relative bias between the two species. The $t^{-1/3}$ behavior, in particular, has so far been observed exclusively in systems with hardcore exclusion. Here we construct a model of particles without hardcore exclusion that reproduces this same $t^{-1/3}$ scaling. In our model, both species undergo asymmetric diffusion with a rate that depends nonlinearly on the local density, a form of transport that, in the absence of the other species, is superdiffusive and falls into the Kardar-Parisi-Zhang universality class. The scaling behavior of $t^{-1/3}$ occurs when at least one of the two species undergoes asymmetric diffusion induced by microscopic processes involving pairs of particles, while asymmetric processes involving groups of three particles lead to $t^{-1/4}$ scaling associated with symmetric diffusion. This demonstrates that the pair annihilation density decay exponent is not determined exclusively by the transport properties of the isolated particle species.

Thermal escape is governed by the Arrhenius law, where the mean escape time scales exponentially with the barrier height. We show that the non-Markovianity induced by time-delayed feedback in the confining force removes this exponential scaling. Beyond a threshold set by the curvature of the minimum, the delay destabilizes the well, and the thermal noise seeds an instability that is subsequently amplified deterministically to the boundary leading to slingshot escape trajectories. The escape time becomes logarithmic in the barrier, its fluctuations follow a Gumbel law, and an optimal delay enables escape faster than free diffusion. Our results propose time delay as a tunable and experimentally feasible control parameter for accelerating activated processes.

Conventionally, the differential form of the Clausius inequality is adopted for infinitesimal irreversible processes. We show that the correct first-order form for infinitesimal quasi-static irreversible processes is the equality d'Q=TdS rather than d'Q < TdS. This provides a purely thermodynamic proof that internal entropy production is second-order or higher and confirms that entropy remains a well-defined state quantity even when quasi-static irreversible processes are included, without resorting to the Boltzmann relation.

The chemical potentials of charged multi-component mixtures are central to electrolyte thermodynamics, but remain difficult to compute from atomistic simulations. The S0 method enables computing chemical potentials of mixtures from equilibrium molecular dynamics simulations. Here we extend the S0 method to charged mixtures by combining the composition-space framework developed in Part I: Neutral Multi-component Mixtures with a Coulombic treatment of the small-wavenumber limits of static structure factors. This approach separates thermodynamically relevant neutral composition fluctuations from forbidden macroscopic charge fluctuations, and further accounts for charge-neutrality constraints. We use the method to compute the chemical potentials of multiple-halide aqueous salt solutions, elucidating the ion-specific thermodynamic effects. We also calculate the mixing free energies of molten salt mixtures, demonstrating the importance of correctly describing long-wavelength electrostatic correlations.

We investigate directed transport in a one-dimensional Brownian flashing ratchet with a piecewise-linear asymmetric periodic potential. Numerical solutions of the Fokker–Planck equation and Brownian dynamics simulations reveal a nonmonotonic dependence of the stationary current on the switching frequency, with a resonant maximum whose position depends on the potential asymmetry $\delta$ and barrier height $\Delta$. For moderate asymmetry, $|\delta|<0.5$, the current obeys a scaling form that separates the dependence on the potential parameters from a common frequency dependence, resulting in a data collapse upon appropriate scaling of the current and frequency. The current amplitude varies linearly with $\delta$, while the resonance frequency follows $\nu(\delta,\Delta)=\nu_0(\Delta)/ [1-b_0\, \delta^2]$. We interpret the resulting $(1-\delta^2)^{-1}$ scaling in terms of coupled relaxation along the two branches of the asymmetric potential, which provides a physical basis for the observed dependence of the resonant frequency on the potential asymmetry.

We prove that the stationary density fluctuation field of the one-dimensional weakly asymmetric simple inclusion process converges to the energy solution of the stochastic Burgers equation. The key technical ingredient is a second-order Boltzmann-Gibbs principle tailored to the microscopic current. As a consequence, we also obtain Ornstein-Uhlenbeck fluctuations for the symmetric inclusion process in a weakly condensing regime, a result of independent interest.

We determine the exact entanglement structure of the conditional state obtained by measuring $m$ corresponding rungs ($m\leq N/2$) in two identical copies of an arbitrary half-filled free fermion Gaussian state and post-selecting the same normalized single-fermion state ($|\psi\rangle = u|10\rangle+v|01\rangle$, where 0 and 1 denote the fermion occupancy on each sites) on each rung. For $m<N/2$, the conditional wavefunction generally depends on the initial state. Nevertheless, whenever the selected outcome has nonzero probability, the state on the unmeasured sites remains Gaussian and factorizes exactly into $N-m$ different orthogonal modes including $m$ inter-copy entangling modes and $N-2m$ spectator modes localized in one copy. Consequently, the entanglement entropy between the unmeasured parts of the two copies is $S=m h_2(|v|^2)$, independent of the initial state, where $h_2(x)=-x\ln x-(1-x)\ln(1-x)$. The success probability is given by $P_m=\det C_L\det(I_m-C_L)=\det(C_{LR}C_{RL})$, determined solely by the initial correlations and independent of ($u$,$v$). Equal-weight Bell post-selection serves as a special case that achieves the maximal entanglement swapping.

Using state-of-the-art numerical techniques, we investigate how quantum phase fluctuations and quasiparticle tunneling shape the behavior of a small-capacitance Josephson junction across Ohmic, sub-Ohmic, and super-Ohmic dissipation regimes. We show that increasing the Ohmic dissipation strength drives a Berezinskii-Kosterlitz-Thouless quantum phase transition at thermodynamic equilibrium. Deviations from Ohmic behavior profoundly alter this scenario: the super-Ohmic regime exhibits no phase transition, whereas the sub-Ohmic regime displays a continuous second-order transition, consistent with the universality classes of the spin-boson model. Within the Ohmic regime, real-frequency linear-response calculations reveal that the phase particle does not undergo the commonly assumed diffusive-to-localized crossover. Instead, finite resistance progressively suppresses the singular zero-frequency response, producing a ballistic-to-localized change in the dynamics. At finite frequencies, coupling to the environment generates a long-lived excitation in the charge response, which evolves into a resonance as the subgap and shunt resistances are reduced.

Can finite-range correlations hide in the statistics of an extended quantum state? The Schrödinger-Nagasawa transform represents the wave function by positive forward and backward diffusion fields whose product is the Born density. We promote them to branching superprocesses, \(\Phi_F\) and \(\Phi_B\): diffusion samples stochastic paths, whereas Bohm/Fisher-controlled branching generates genealogies of alternative continuations. With rates evaluated on the prescribed Born density, their means reproduce Schrödinger dynamics exactly. The connected sector exhibits supercritical, critical, and subcritical clustering in confinement and has critical dimension \(d_c=2\) in free space. Stationary eigenmodes remain extended; subcritical clusters acquire the reduced de~Broglie scale. We then let the branching rate respond to the fluctuating product \(\Phi_F\Phi_B\). In the reciprocal basis, the fields equal a smooth reference \(R\) plus centered fluctuations \(\psi_F,\psi_B\), defining the signed kernel \(C_{\rm FB}(x,y)=\mathbb E_\omega[\psi_F(x)\psi_B(y)]\). On the anticorrelated branch, \(\rho_{\rm BSM}(x)=-C_{\rm FB}(x,x)\) is the positive organized connected weight. Stationary matching of this weight to the Born profile is imposed as a closure condition, while off-diagonal decay defines the correlation range. The pair equation splits into collective and relative sectors: spectral cancellation selects the Born collective mode, while suppression of the leading density fluctuation selects the anticorrelated source channel. With relative diffusivity \(D_{\rm eff}\) and positive relaxation rate \(\mu_{\rm FB}\), correlations have screening length \(\xi_{\rm FB}=\sqrt{D_{\rm eff}/\mu_{\rm FB}}\). Within this reciprocal closure, an extended Born-shaped connected weight and a finite correlation range can coexist in one stochastic kernel.

Trajectories in complex systems often contain structure that is not captured by the population-averaged, scalar statistics traditionally used to describe them. Here we reconstruct, through independent numerical re-simulation rather than literature review, a systematic search across representational forms - from single scalar statistics, through two independent discrete multi-feature classification attempts, to a trajectory-level structural representation and a diagnostic surrogate test, to a continuous low-dimensional compression - applied to the same underlying trajectory ensemble in a critical complex system. Predictive failure is not explained by insufficient correlation: one discrete representation's own strongest individual feature correlates with the target more strongly ($r=0.588$) than the eventual successful representation's own headline statistic ($r=0.540$), yet fails to separate the ensemble's two most consequential cases - while the continuous representation succeeds, robustly, across eight independent scale-removal folds ($r=0.677$-$0.717$, all $p<0.001$). The determining factor was not correlation strength but whether the representation preserved the structure relevant to prediction. These results indicate that the choice of representation determines whether predictive structure in complex systems becomes accessible - a scientific variable in its own right, not a downstream analysis choice.

The capacity for autonomous structural reconfiguration is a defining trait of living systems, yet it remains elusive in artificial active matter. Here, we report the discovery of active embracement, a non-equilibrium phenomenon where active star polymers, comprising a central core and self-propelled monomeric arms, transition from open configurations to tightly collapsed, “hugging” states. By combining polymer experiments using connected vibrobots with simulations, we demonstrate that internal self-propulsion fundamentally overrides the steric repulsion that keeps passive polymers dispersed. This active drive enables a suite of behaviors unattainable in equilibrium systems: individual star polymers undergo a globular-like self-collapse, multiple star polymers mutually intertwine and mutually embrace, and can spontaneously embrace and capture surrounding passive particles. Our findings reveal that active embracement is a distinct kinetic phase that allows star polymers to function as autonomous tweezers. By bridging the gap between macroscopic robotic collectives and microscopic polymer physics, this work provides a versatile blueprint for the design of smart materials capable of targeted cargo capture and self-directed assembly in complex environments.

We study the phase diagram and critical properties of the $q = 3$ and $q = 8$ Potts models with spatially correlated disorder governed by a power-law decay with exponent $a$. Building on the phase diagram proposed by Chippari et al. [3], where a transition from finite-disorder to infinite-disorder fixed points was identified as a function of a, we refine this picture by using wrapping probabilities of Fortuin-Kasteleyn clusters. Furthermore, using magnetic observables, we clarify the physical origin of the double-peak structure in the magnetic susceptibility and establish the validity of the hyper-scaling relation across all investigated regimes.