We study stochastic transport of interacting particles on a disordered network described by the random comb geometry. The model is defined on a one-dimensional backbone from which branches of random lengths emanate, providing a minimal model of percolation networks beyond the critical percolation probability. The dynamics obeys local detailed balance with respect to a Bose-Hubbard Hamiltonian containing both an external bias and on-site repulsion. This choice yields an analytically tractable steady state through a mapping to the zero-range-process. We compute the backbone current, branch density profiles, and macroscopic drift velocity, and analyze how bias and interactions compete to shape transport. The backbone current increases monotonically with density, while the drift velocity displays a non-monotonic dependence on the external field, remaining finite for any nonzero bias, in contrast to the vanishing drift velocity of noninteracting particles beyond a threshold bias. Density profiles along branches exhibit stepwise plateaus governed by the ratio of interaction to bias energy. These results highlight how repulsive interactions suppress trapping and restore transport in disordered geometries, bridging earlier studies of field induced drift in random networks with the physics of disordered Bose-Hubbard systems.

Driven nonequilibrium lattice models have wide-ranging applications in contexts such as mass transport, traffic flow, and transport in biological systems. In this work, we investigate the steady-state properties of a one-dimensional lattice system that allows multiple particle occupancy on each site. The particles undergo stochastic nearest-neighbor jumps influenced by both a directional bias and on-site repulsive interactions of the Bose-Hubbard type. With periodic boundary conditions, we observe a nonmonotonic dependence of intersite correlation functions on the interaction strength. At large interaction strengths, the state consists of quiescent stacks of stationary particles along with an emergent asymmetric simple exclusion process, and the particle current exhibits a periodic dependence on density. In contrast, with open boundary conditions, the system displays steplike density profiles reminiscent of those in tilted Bose-Hubbard systems, and a regime with a macroscopic number of empty sites followed by a steep parameter-dependent increase in density. Our results highlight how the interplay between drive, interaction, and boundary conditions leads to distinctive signatures on the current and density profiles in the steady state in different regimes.

The Light-Heavy (LH) model involves two species of particles (light and heavy) coupled with a fluctuating surface (described by tilts). The dynamics include the inherent diffusion of the particles (or tilts) as well as the drive provided by the tilts (or particles). When the two are of similar magnitude, the system lies in the unscaled (uLH) regime, while a significantly weaker drive leads to the scaled (sLH) regime. In the unscaled limit, the model exhibits an order-disorder transition characterized by the fluctuation-dominated phase ordering (FDPO). In this state, interestingly the dynamics is driven by multiple modes, giving rise to dynamic clusters. Away from the critical regime the disordered phase retains vestiges of FDPO behavior on length scales smaller than the correlation length. We examine this local FDPO-like behavior by using a scaling function that links the off-critical and critical regimes. We next turn to the scaled model and show that the multi-mode dynamics present in the unscaled regime is replaced by dynamics that is effectively controlled by a single dominant mode in the scaled regime. Concurrently, the two-point correlations change from the 𝒪(1) FDPO form to an anomalous long-range form that decays as 1/√(L). Drawing on the analogy with the sABC model, where similar anomalous correlations appear at criticality, we derive an analytical expression for the two-point correlation function using the same approach used for that model.

As the simplest model of transport of interacting particles in a disordered medium, we consider the asymmetric simple exclusion process (ASEP) in which particles with hard-core interactions perform biased random walks, on the supercritical percolation cluster. In this process, the long time trajectory of a marked particle consists of steps on the backbone, punctuated by time spent in side branches. We study the probability distribution in the steady state of the waiting time T_{w} of a randomly chosen particle, in a side branch since its last step along the backbone. Exact numerical evaluation of this on a single side branch of length L=1 to 9 shows that for large fields, the probability distribution of logT_{w} has multiple well separated peaks. We extend this result to a regular comb, and to the ASEP on the percolation cluster. We show that in the steady state, the fractional number of particles that have been in the same side branch for a time interval greater than T_{w} varies as exp(-csqrt[logT_{w}]) for large T_{w}, where c depends only on the bias field. However, these long timescales are not reflected in the eigenvalue spectrum of the Markov evolution matrix. The system shows dynamical heterogeneity, with particles segregating into pockets of high and low mobilities.

We introduce and study a non-equilibrium stochastic model of two fluctuating interfaces which interact through short-range attractive interactions at their points of contact. Beginning from an entangled state, the system exhibits diverse dynamics – ranging from fast transients with small lifetimes to ultraslow evolution through quasi-stationary states – and reaches stuck, entangled, or detached steady states. Near the stuck-detached transition, two distinct dynamical modes of evolution co-occur. When the two surfaces evolve through similar dynamics (both Edwards-Wilkinson or both Kardar-Parisi-Zhang), the invariant measure is determined and found to have an inhomogeneous product form. This exact steady state is shown to be the measure of the equilibrium Poland-Scheraga model of DNA denaturation.

Many physical systems, including some examples of active matter, granular assemblies, and biological systems, show fluctuation-dominated phase ordering (FDPO), where macroscopic fluctuations coexist with long-range order. Most of these systems are out of equilibrium. By contrast, a recent work has analytically demonstrated that an equilibrium one-dimensional truncated inverse distance square Ising (TIDSI) model shows FDPO. The analytical results rely on a cluster representation of the model that we term TIDSI-CL and are governed by the ratio, c, of the long-range interaction strength to the critical temperature. We show that the allowed range of c is very narrow in the original TIDSI model while it is unbounded in TIDSI-CL. We perform Monte Carlo simulations for the TIDSI model and show consistency with the analytical results in the allowed range of c. The correlation length grows strongly on approaching the critical point, leading to a broad near-critical region. Within this region, α, which is the cusp exponent of the power-law decay of the scaled correlation function at criticality, changes to α^{eff}. We also investigate the coarsening dynamics of the model: The correlation function, domain size distribution, and aging behavior are consistent with the equilibrium properties upon replacing the system size, L, by the coarsening length, L(t). The mean largest cluster size shows logarithmic corrections due to finite L and waiting time, t_{w}. The aging autocorrelation function exhibits two different scaling forms, characterized by exponents β and γ, at short and long times compared to t_{w}, where β=α/2.

We study the exact fluctuating hydrodynamics of the scaled Light-Heavy model (sLH), in which two species of particles (light and heavy) interact with a fluctuating surface. This model is similar in definition to the unscaled Light-Heavy model (uLH), except it uses rates scaled with the system size. The consequence, it turns out, is a phase diagram that differs from that of the unscaled model. We derive the fluctuating hydrodynamics for this model using an action formalism involving the construction of path integrals for the probability of different states that give the complete macroscopic picture starting from the microscopic one. This is then used to obtain the two-point steady-state (static) correlation functions between fluctuations in the two density fields in the homogeneous phase. We show that these theoretical results match well with microscopic simulations away from the critical line. We derive an exponentially decaying form for the two-point steady-state correlation function with a correlation length that diverges as the critical line is approached. Finally, we also compute the dynamic correlations in the homogeneous phase and use them to determine the relaxation dynamics as well as the dynamic exponents of the system.

Large-mass condensates, which coexist with a power-law-decaying distribution in the one-dimensional Takayasu model of mass aggregation with input, were recently found in numerical simulations. Here, we establish the occurrence of condensates by analyzing exact recursions for finite systems and further show that they have a strong effect on the properties of the system. In the steady state of a large but finite system, there is a single condensate, whose random movement through the system leads to a reorganization of the mass profile on a macroscopic scale. A scaling analysis of the mean mass and standard deviation at different distances from the condensate leads to the surprising conclusion that the mass distribution on a macroscopic number of sites around the condensate follows a power-law decay with an exponent 5/3, while farther-away sites show the customary Takayasu exponent 4/3, with a crossover in between. Finally, the exit of condensates from a system with open boundaries has a strong effect on the temporal fluctuations of the total mass in the steady state. Their departure is followed by a buildup of mass and subsequent departures, leading to strong intermittency, established through a divergence of the flatness as the scaled time approaches zero.

This text was supposed to be included in the book "50 years of the renormalization group, Dedicated to the Memory of Michael E. Fisher", edited by A. Aharony, O. Entin-Wohlman, D. Huse and L. Radzihovsky, World Scientific, Singapore (2024). It will be included in future printings and in the electronic version of the book.

We determine the arrangement of spins in the ground state of the XY model with quenched, random fields, on a fully connected graph. Two types of disordered fields are considered, namely randomly oriented magnetic fields, and randomly oriented crystal fields. Orientations are chosen from a uniformly isotropic distribution, but disorder fluctuations in each realization of a finite system lead to a breaking of rotational symmetry. The result is an interesting pattern of spin orientations, found by solving a system of coupled, nonlinear equations within perturbation theory and also by exact numerical continuation. All spins lie within a cone for small enough ratio of field to coupling strength, with an interesting distribution of spin orientations, with peaks at the cone edges. The orientation of the cone depends strongly on the realization of disorder, but the opening angle does not. In the case of random magnetic fields, the cone angle widens as the ratio increases till a critical value at which there is a first order phase transition and the cone disappears. With random crystal fields, there is no phase transition and the cone angle approaches 180^∘ for large values of the ratio. At finite low temperatures, Monte-Carlo simulations show that the formation of a cone and its subsequent alignment along the equilibrium direction occur on two different time scales.

The autocorrelation function in many complex systems shows a crossover in the form of its decay: from a stretched exponential relaxation (SER) at short times to a power law at long times. Studies of the mechanisms leading to such multiple relaxation patterns are rare. Additionally, the inherent complexity of these systems makes it hard to understand the underlying mechanism leading to the crossover. Here we develop a simple one-dimensional spin model, which we call a domain wall (DW) to doublon model, that shows such a crossover as the nature of the excitations governing the relaxation dynamics changes with temperature and time. The relevant excitations are DWs and bound pairs of DWs, which we term 'doublons'. The diffusive motion of the DWs governs the relaxation at short times, whereas the diffusive motion of the doublons yields the long-time decay. This change of excitations and their relaxation leads to a crossover from SER to a power law in the decay pattern of the autocorrelation function. We augment our numerical results with simple physical arguments and analytic derivations.

Fluctuation-dominated phase ordering refers to a steady state in which the magnitude of long-range order varies strongly owing to fluctuations, and to the associated coarsening phenomena during the approach to steady state. Strong fluctuations can lead to a number of interesting phenomena, including a cusp singularity in the scaled correlation function, implying the breakdown of the Porod Law. First identified in a nonequilibrium system of passively sliding particles on a fluctuating surface, fluctuation-dominated order also occurs in several other systems, including an equilibrium Ising model with long-range interactions. This article discusses these systems, and others where clustering effects are stronger.

We use extreme value statistics to study the dynamics of coarsening in aggregation-fragmentation models which form condensates in the steady state. The dynamics is dominated by the formation of local condensates on a coarsening length scale which grows in time in both the zero range process and conserved mass aggregation model. The local condensate mass distribution exhibits scaling, which implies anomalously large fluctuations, with mean and standard deviation both proportional to the coarsening length. Remarkably, the state of the system during coarsening is governed not by the steady state, but rather a preasymptotic state in which the condensate mass fluctuates strongly.

The autocorrelation function in many complex systems shows a crossover in the form of its decay: from stretched exponential relaxation (SER) at short times to power law at long times. Studies of the mechanisms leading to such multiple relaxation patterns are rare. Additionally, the inherent complexity of these systems makes it hard to understand the underlying mechanism leading to the crossover. Here we develop a simple one-dimensional spin model, which we call a Domain Wall (DW) to Doublon model, that shows such a crossover as the nature of the excitations governing the relaxation dynamics changes with temperature and time. The relevant excitations are DWs and bound pairs of DWs, which we term `doublons'. The diffusive motion of the DWs govern the relaxation at short times, whereas the diffusive motion of the doublons yields the long time decay. This change of excitations and their relaxation leads to a crossover from SER to power law in the decay pattern of the autocorrelation function. We augment our numerical results with simple physical arguments and analytic derivations.

Models of particles driven by a one-dimensional fluctuating surface are known to exhibit fluctuation dominated phase ordering (FDPO), in which both the order and fluctuations appear on macroscopic scales. Highly dynamic and macroscopically broad interfacial regions, each composed of many domain walls, appear between macroscopically ordered regions and consequently the scaled correlation function violates the Porod law. We focus on two essential quantities which together quantify the unique characteristics of FDPO, namely the total number of domain walls and the length of the largest ordered domain. We present results in the context of coarse-grained depth (CD) models, both in steady state and while coarsening. Analytic arguments supported by numerical simulations show that even though domain wall number fluctuations are very strong, the associated variance remains constant in time during coarsening. Further, the length of the largest cluster grows as a power law with significant multiplicative logarithms which involve both the time and system size. In addition, we identify corrections to the leading power law scaling in several quantities in the coarsening regime. We also study a generalisation of the CD model in which the domain wall density is controlled by a fugacity and show that it maps on to the truncated inverse distance squared Ising model. The generalised model shows a mixed order phase transition, with the regular CD model (which exhibits FDPO) corresponding to the critical point.

Integrin is an important transmembrane receptor protein which remodels the actin network and anchors the cell membrane towards the extracellular matrix via mechanochemical pathways. The clustering of specific lipids and lipid-anchored proteins, which is essential for a certain type of endocytosis process, is facilitated at integrin-mediated active regions. To study this, we propose a minimal exactly solvable model which includes the interplay of stochastic shuttling between integrin on and off states with the intrinsic dynamics of the membrane. We propose a two-step mechanism in which the integrin induces an aster-like arrangement in the actin network, followed by clustering of lipids in that region. We obtain an analytic expression for the deformation and local membrane velocity, and thereby the evolution of clustering mediated by a single integrin. The deformation evolves nonmonotonically and its dependence on the stochastic shuttling timescales and membrane properties is elucidated. Our estimates of the area of the deformed region and the number of lipids in it indicate strong clustering.

The Takayasu aggregation model is a paradigmatic model of aggregation with mass injection, known to exhibit a power law distribution of mass over a range which grows in time. Working in one dimension we find that the mass profile in addition shows distinctive dynamic condensates which collectively hold a substantial portion of the mass (approximately 80% when injection and diffusion rates are equal) and lead to a substantial hump in the scaled distribution. To track these, we monitor the largest mass within a growing coarsening length. An interesting outcome of extremal statistics is that the mean of the globally largest mass in a finite system grows as a power law in time, modulated by strong multiplicative logarithms in both time and system size. At very long times, in a finite system, the state consists of a power-law-distributed background with a condensate whose mass increases linearly with time.

We study the competition between field-induced transport and trapping in a disordered medium by studying biased random walks on random combs and the bond-diluted Bethe lattice above the percolation threshold. While it is known that the drift velocity vanishes above a critical threshold, here our focus is on fluctuations, characterized by the variance of the transit times. On the random comb, the variance is calculated exactly for a given realization of disorder using a 'forward transport' limit which prohibits backward movement along the backbone but allows an arbitrary number of excursions into random-length branches. The disorder-averaged variance diverges at an earlier threshold of the bias, implying a regime of anomalous fluctuations, although the velocity is nonzero. Our results are verified numerically using a Monte Carlo procedure that is adapted to account for ultra-slow returns from long branches. On the Bethe lattice, we derive an upper bound for the critical threshold bias for anomalous fluctuations of the mean transit time averaged over disorder realizations. Finally, as for the passage to the vanishing velocity regime, it is shown that the transition to the anomalous fluctuation regime can change from continuous to first order depending on the distribution of branch lengths.

We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the lattice in one and two dimensions and derive exact results in the continuum limit. Next, we compute the large deviation free-energy function in both one and two dimensions, which we use to compute the moments and the cumulants of the displacements exactly at late times. Our exact results demonstrate that the cross-correlations between the motion in the x and y directions in two dimensions persist in the large deviation function. We also demonstrate that the large deviation function of an active particle with diffusion displays two regimes, with differing diffusive behaviors. We verify our analytic results with kinetic Monte Carlo simulations of an active lattice walker in one and two dimensions.

We use large deviation theory to obtain the free energy of the XY model on a fully connected graph on each site of which there is a randomly oriented field of magnitude h . The phase diagram is obtained for two symmetric distributions of the random orientations: (a) a uniform distribution and (b) a distribution with cubic symmetry. In both cases, the disorder–averaged ordered state reflects the symmetry of the underlying distribution. The phase boundary has a multicritical point (MCP) which separates a locus of continuous transitions (for small values of h ) from a locus of first order transitions (for large h ). The free energy is a function of a single variable in case (a) and a function of two variables in case (b), leading to different characters of the MCPs in the two cases. We find that the locus of continuous transitions is given by the same equation for a family of quadriperiodic distributions, which includes the distributions (a) and (b). However, the location of the MCP and the nature of ordered state depend on the form of the distribution. The disorder-averaged ground state energy is found exactly, and the specific heat is shown to approach a constant as temperature approaches zero.