Papers

We consider the simultaneous estimation of multiple parameters in Hamiltonians of the form $H_{\vec{\theta}}= H_{\vec{\theta}}^P + H^C$, where $H_{\vec{\theta}}^P$ encodes the unknown parameters and $H^C$ is a control term. We consider three basic paradigmatic frameworks for metrology: (i) dynamical, where the parameters are encoded unitarily, (ii) thermal, and (iii) ground state. We generalize previous bounds for single-parameter estimation to simultaneous multiparameter estimation for all three metrological settings and establish the conditions required for their saturability. We then focus on vector magnetometry, deriving tight bounds for the simultaneous estimation of magnetic-field components and identifying optimal many-body control Hamiltonians $H^C$ that saturate them. These results establish the fundamental limits of vector magnetometry for unitary dynamics, thermal equilibrium states, and ground states.

The probability that several independent random walks avoid one another decays algebraically with the number of steps $N\to\infty$ in two and three dimensions, $P_N\sim N^{-\xi/2}$, where the intersection exponent $\xi$ depends on the number of random walks. More generally, one may consider several groups of independent random walks, with intersections allowed within each group and forbidden between different groups. We first study intersection exponents in two dimensions, where our results agree with the known exact formulas and test the numerical approach. In three dimensions, where no general exact expression is known, we determine $\xi$ for a range of cases. For two groups containing $k$ and $m$ random walks, respectively, we obtain the exponents $\xi(k,m)$ for $k=1,2$ and $m=1,2,\ldots,7$. We further extend $m$ to a continuous parameter $\lambda$, allowing us to obtain numerical values for $\xi(k,\lambda)$ for $k=1,2,3$ and $0.25\leq\lambda\leq3$. Away from the smallest moment orders, these functions increase with $\lambda$ with decreasing slopes, as expected from the strict concavity of Brownian intersection exponents. We also investigate three-group configurations for several representative cases. The resulting integer and continuous exponents supply numerical values for analytical studies of non-intersecting random paths.

We identify a broad class of communication channels that captures common physical constraints in both artificial and natural systems and derive bounds on their thermodynamic cost. We find that the entropy production per channel use is bounded from below by the input-output mutual information, and their ratio -mutual information divided by entropy production- defines the thermodynamic efficiency. Unlike previous studies of energy-constrained communication channels, our analysis shows that thermodynamic costs must be assigned not only to the input symbols themselves, but also to transitions between successive symbols. As a result, maximizing thermodynamic efficiency favors a biased input that switches only rarely, rather than the capacity-achieving input. For the binary symmetric channel, this preference emerges through a pitchfork bifurcation that spontaneously breaks the symmetry of the channel. A minimal model of cellular sensing exhibits the same phenomenon.

Intrigued by the response of T cell receptors to the presence of a few agonist ligands, we propose a minimal model that can achieve similar performance. The model consists of a small cluster of immobile receptors that bind reversibly to two types (correct/incorrect) of ligands in the environment, with slightly weaker binding strength for the incorrect one. It features binding-state coupling between nearest-neighbor receptors, and receptors in the bound/free states are activated/deactivated by specific enzymes, with rates that allow kinetic proofreading. It is found that, for a range of binding-state coupling strength, incorrect ligands alone cannot activate the receptors, but the binding of merely one correct ligand to a receptor is sufficient to promote the activation of other receptors via induced binding to incorrect ligands. Both response time and signal amplification increase as the receptor binding-state coupling strength increases until it reaches an optimal range to achieve the most rapid and sensitive response. These results suggest a possible mechanism for a speedy and specific response of receptors to very few correct ligands in biological and artificial systems at the subcellular scale.

I connect two literatures developed independently: the path-measure formulation of non-equilibrium statistical mechanics, where a trajectory action decomposes into a time-antisymmetric (entropic) and time-symmetric (frenetic) sector, and the stochastic modelling of ribosomal traffic on messenger RNA. The biological target is a proposed intervention – antisense oligonucleotide (ASO) interference with wobble-uridine (U34) modification of transfer RNA – whose intended effect is not to abolish translation but to perturb its timing, driving ribosome collisions and collision-triggered transcript decay preferentially on high-flux, codon-biased transcripts. Dynamical-activity and large-deviation analyses of generic lattice exclusion models – notably the symmetric and totally asymmetric simple exclusion processes – are well established. To my knowledge their formalization specifically for ribosomal traffic queues, translation elongation, and collision-triggered no-go decay remains unoccupied; this paper addresses that narrower gap, not the general one. Two claims here are load-bearing and untested. First, selectivity: transcripts whose loss is therapeutically desirable are separable, by vulnerable-codon-pair burden, from transcripts whose loss is toxic. Second, non-redundancy: the frenetic decomposition yields predictions, specific to ribosomal queueing and collision-triggered decay, not already obtainable from rate-level exclusion-process models or existing activity/large-deviation analyses of exclusion processes. This paper establishes neither. It specifies both as falsifiable tests with pre-registered decision rules, including outcomes under which the framework should be abandoned or narrowed. It is a research programme proposal, not a result. No new experimental, computational, or bioinformatic results are reported.

We prove that the subcritical mass gap is comparable to the Lee-Yang gap over free-boundary square boxes raised to the power $8/15$. Random-current moment bounds, a correlation inequality and a finite block expansion give the upper bound. The lower bound follows from the Lee-Yang product, the susceptibility and the critical magnetization. We also obtain uniform factorial bounds on the field derivatives of the pressure.

Generally the mobility in the Cahn-Hilliard equation may depend on the order parameter; this was evident already from the original derivation. However, explicit introduction of such dependence results in tremendous calculation difficulties, so the constant mobility approximation was commonly used. On the other hand, in several cases this dependence appeared to be crucial, so more realistic expressions were applied, usually positive powers or polynomials; still, for such dependencies only approximate and/or numerical solution exist. We consider 'reciprocal' linear and quadratic dependencies of mobility on the order parameter; for these dependencies exact traveling wave solutions are obtained. Even more, in a limited interval of the parameters reciprocal quadratic mobilities are rather good approximations to polynomial mobilities.

We characterise the approach to nonequilibrium steady state in nested resetting processes by deriving accumulation times for the system, which quantify the effective first-passage time for the local establishment of steady state. For equal resetting rates, we obtain closed-form expressions showing that relaxation propagates as a wavefront with a delay that increases linearly along the resetting chain. We then extend this analysis to heterogeneous resetting rates, deriving exact steady-state distributions, spatial moments and accumulation times for both degenerate and non-degenerate resetting rates. We show that relaxation to steady state is ultimately governed by the minimum resetting rate in the system and, in degenerate systems, its multiplicity. These results provide a comprehensive analytical description of first-passage to nonequilibrium steady state in hierarchically coupled stochastic resetting systems, with potential applications to relaxation and transport processes in biological systems.

Whether a nonequilibrium quantum-critical steady state can be described by an effective temperature remains an open question. We address it in a voltage-biased electronic Lipkin–Meshkov–Glick model, a minimal driven-dissipative model of a collective spin coupled to metallic leads. A controlled large-$N$ treatment reveals an overdamped open quantum-critical regime distinct from the closed model. The surrounding quantum critical fan, organized by temperature and voltage, is explored and fluctuations are found to be governed by an effective temperature $T_{\rm eff}$. At strong bias, the transition becomes first-order through a tricritical point. Remarkably, the same $T_{\rm eff}$ governs the strongly driven regime, but now varies across the entire order-parameter landscape. The steady state is therefore selected by a non-Boltzmann rule, shifting the first-order transition away from the equal-depth point of a deterministic potential. Driven criticality thus remains organized by an effective temperature while revealing non-thermal state selection.

We introduce and develop Functional Dimensional Regularization (FDR), a novel functional renormalization group (RG) framework that extends the principles of dimensional regularization beyond the perturbative $\varepsilon$-expansion. The key insight is to define the FDR beta functions, valid in any continuous dimension $d$ and characterized by threshold functions, as a sum over all critical dimensions of the corresponding scheme-independent beta functions computed in DR. We demonstrate that this formalism exhibits all the hallmarks of a fully-fledged functional RG. Within the local potential approximation, we perform a general fixed point analysis, recovering the $\varepsilon$-expansion for all multi-critical models and establishing a direct connection to functional scaling solutions, spike plots, and eigen-perturbation spectra. Extending the framework to the second order of the derivative expansion, we compute the critical exponents for the Ising universality class in $d=3$, providing estimates for the anomalous dimension $\eta$ and RG spectrum that converge rapidly and show favorable agreement with state-of-the-art results. We present a detailed comparison with non-perturbative and proper-time RG approaches, highlighting structural similarities and key differences. In particular, we introduce a “distillation” procedure that allows one to systematically derive the FDR flow from any other one-loop-exact RG scheme. Our results establish FDR as a self-contained and competitive RG framework that combines the technical simplicity of dimensional regularization with the versatility of the functional approach.

We present bulk-primary operator-product coefficients for every Virasoro minimal model with an $E_6$, $E_7$, or $E_8$ modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group $6j$ symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices $s$, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic $s=1$ seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for $E_6$, $E_7$, and $E_8$, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).

Fast, high-fidelity qubit initialization is a key primitive for scalable quantum information processing, but conventional reset protocols either require long relaxation times or discard information contained in continuous measurement records. We demonstrate an adaptive measurement-based reset protocol for superconducting qubits that uses Bayesian inference to update the qubit-state estimate after each readout and dynamically adjust the feedback threshold. Unlike fixed-threshold or repeat-until-success (RUS) protocols, the method utilizes the full analog measurement history, enabling the reset decision to become progressively more conservative as confidence in ground-state preparation increases. Implemented with real-time FPGA-based feedback on a dispersively readout transmon qubit, the protocol achieves a ground-state initialization fidelity of $99.44\pm0.04\%$ after a small number of reset rounds. We further compare adaptive reset with RUS strategies in multi-qubit experiments and show that adaptive thresholding provides deterministic reset duration while maintaining a five-qubit simultaneous-reset fidelity of $98.78 \pm 0.19\%$. These results establish Bayesian adaptive thresholding as a practical and scalable route to fast qubit reset in superconducting quantum processors.

Can primordial cosmological correlations originate from statistical fluctuations of an early-universe thermal system? We connect grand-canonical thermodynamics, stochastic transport, and gauge-invariant cosmological perturbations for a charged fluid at finite chemical potential. Energy-charge susceptibilities determine curvature, charge-isocurvature, and cross-correlation amplitudes, while transport sets when each mode leaves equilibrium. For an isolated, adiabatic conformal plasma with conserved charge-to-entropy ratio and power-law diffusion, diffusive freeze-out yields the universal spectrum $\mathcal P_S(k)\propto k^3$, with $n_{\rm iso}=4$, while equal-time local-equilibrium curvature-isocurvature covariance vanishes in the conformal source basis. Thus neither finite chemical potential nor replacing Hubble crossing by diffusion crossing alone yields approximate scale invariance. We evade this obstruction with an open subsystem of massless Dirac fermions during a sourced quasi-de Sitter phase. There, grand-canonical energy cumulants generate a nearly scale-invariant red curvature spectrum. A benchmark gives $\mathcal P_{\zeta_X}(k_\star)=2.10\times10^{-9}$ and $n_s(k_\star)=0.9649$, stays within $0.30%$ of the fitted power law over $0.002\leq k/k_\star\leq4$, and predicts weak negative running and small positive intrinsic non-Gaussianity from higher-order cumulants, $f_{\rm NL}(k_\star)\simeq0.106$ and $g_{\rm NL}(k_\star)\simeq0.0158$. With a two-helicity vacuum tensor spectrum, the tensor-to-scalar ratio is $r_{t/s}(k_\star)=1.82\times10^{-4}$. This isolates the limitation of isolated conformal thermal seeding and the open-system ingredient needed to overcome it. Thermal energy-charge fluctuations thus provide a calculable source of realistic scalar correlations, while reservoir dynamics, equilibration, and later curvature-isocurvature transfer must come from a microscopic model.

In classical systems, rugged potential energy landscapes provide a transparent mechanism for the celebrated Mpemba effect, in which hotter initial states cool down faster. This picture generally does not survive in quantum systems. Here we show how to design a quantum energy landscape with local dissipation leading to an Mpemba effect with parametrically separated timescales. Our approach uses Stark localization to design an energy landscape and localized incoherent hopping as cooling mechanism. The hops are triggered by rare "detection" events whose rate grows with energy, allowing hotter states to cool faster and producing super-exponentially separated cooling rates for localized initial states. We further show that the effect is dramatically enhanced by collective hopping of bound pairs in the presence of attractive on-site interactions. These findings have clear experimental signatures accessible in present-day setups.

The quantum metric quantifies the total ground-state response along a parameter direction, but does not resolve how excitations share that response. We define channel concentration (CC) as the sum of squared normalized response weights over specified excitation channels, such as momentum blocks. An exact finite-size theorem yields thermodynamic CCs of 2/3 for the field response of the critical transverse-field Ising model and 1/3 for the half-filled XX pairing response, despite the same leading metric scaling. In finite-size approaches to the XY Lifshitz point, field and anisotropy perturbations yield different concentrations despite a common limiting Hamiltonian with quadratic dispersion. In a unitary 1+1-dimensional conformal field theory (CFT) on a circle, we consider a nondegenerate vacuum in a fixed sector perturbed by one spatially integrated scalar primary. We derive complete zero-momentum energy-level response weights, including descendants. For scaling dimension $0<\Delta<3/2$, these weights determine the normalized response distribution and an exact universal concentration function. The expression reproduces the exact Ising lattice limit 2/3 and gives approximately 0.8515 for the three-state Potts thermal field, compared with approximately 0.800 from an exponent-only approximation. Finite-size interacting calculations compare concentrations and ranked response weights over many-body energy levels. These exact benchmarks show which response distinctions total metric scaling misses and guide comparisons with finite-size interacting spectra.

We present a theoretical proposal for measuring correlation functions in a quantum field system based on the use of quantum quenches. Using the examples of massive and massless scalar field theories, we show that a short-term perturbation of the system leads to a dependence of the further evolution of the average field on the initial state's correlation functions. Considering two types of perturbations, namely local and nonlocal quenches, we show which correlation functions can be measured in each of these cases. The proposed procedure can be applied to study non-Gaussian correlations.

We investigate how slow out-of-equilibrium modes (OEM), introduced to extend the regime of validity of hydrodynamics near the critical point, fundamentally reshape the power spectrum of dynamical density fluctuations. We have used the equation of motion of slow modes for the situation when the extensive nature of thermodynamics is not altered due to the introduction of OEM. We find that the extensivity condition puts an extra constraint on the coupling of OEM with the four divergence of velocity. In the absence of OEM, the dynamic structure factor exhibits three Lorentzian peaks, one at zero frequency ($\omega$), called Rayleigh peak and the other two called Brillouin peaks located symmetrically about $\omega=0$. When the system is away from the critical point the inclusion of the OEM preserves the symmetry of the distributions but all the peaks acquire reduced widths and the Brillouin peaks adopt unequal heights. Irrespective of the value of OEM (zero or nonzero) the Brillouin peaks vanish near the critical point. The width of the Rayleigh peak substantially reduced in presence of OEM. Such reductions of the widths in presence of the OEM indicates the reduction in the decay rate of the fluctuation which leads to slowing down of decay of the fluctuations, a distinctive feature of the critical point.

A driven colloidal rotor sustains a circulating current whose fluctuations one would like to bound. Any such bound rests on how strongly the current responds to a perturbation and on how much extra dissipation that perturbation costs, so there is a well-posed question of where to push: the response per unit Onsager-Machlup cost depends on the radius at which the probe acts, and is maximised at a definite one. We answer this for the Brownian gyrator and a quadrupolar shear under a Gaussian envelope, a probe chosen so that linear response is exactly blind to it at every observation window, so that the entire signal is second order. The problem separates: the cost is radial and does not see the circulation, while the response lives in the $m=3$ angular sector, where the resolvent reduces to Kummer's equation with $b=4$ and the susceptibility is a hypergeometric function of the envelope width. Maximising the ratio gives the optimal probing radius in closed form. It is set by whichever of the system's two clocks is faster: for weak driving $r^*=1.1264\sqrt{D/\gamma}$, the thermal radius of the trap, and for strong driving $r^*=1.5563\sqrt{D/\Omega}$, the distance diffused in one radian of rotation, with the trap stiffness dropping out entirely. The crossover is at $\Omega=\gamma$. Both limits, and the prefactors, are confirmed against direct simulation.

We report the first calculation of the lowest-known energy configuration for $N=100\,000$ classical point charges confined to a disk and interacting via the $1/r$ Coulomb potential - a system size never before achieved for the Thomson problem in a disk, more than doubling the previous record of $N=40\,886$ reported by Amore and Zarate. An adaptive defect-targeting subdomain optimization strategy yields an approximately linear growth of the wall-clock time per cycle with $N$; for $N=100\,000$ a single $24$-core CPU workstation without GPU acceleration required $\approx31$ hours. The energy $E_{\min}=7.80466624157\times10^{9}$ deviates from the standard empirical asymptotic expansion by only $2.525\times10^{-7}$. We demonstrate that this remarkable agreement provides a stringent test of the rigorous next-order asymptotic theory for Riesz interactions, while the residual deviation is consistent with an $O(N^{4/3})$ boundary-layer correction whose absence from the standard fitting basis biases the empirical $N^{3/2}$ coefficient away from its exact crystalline value, thereby tying the deviation to the two-dimensional crystallization conjecture. Bond-orientational order analysis reveals a polycrystalline bulk threaded by radial grain boundaries, establishing a new benchmark for two-dimensional Coulomb systems.

The fact that generic isolated many-body quantum systems thermalize is understood using the eigenstate thermalization hypothesis (ETH). In recent years, there has been much interest in the behavior of the ETH spectral functions, which characterize the smooth dependence of the variance of the off-diagonal matrix elements of observables on the associated energy and frequency, and whose low-frequency part contains information about the long-time dynamics. In finite systems described by the ETH, the spectral functions are expected to exhibit plateaus below a characteristic frequency $\omega^{}_{\mathrm{ETH}}$. In this regime, the statistics of the matrix elements of observables are expected to be described by random matrix theory. Related frequencies that have been studied in the literature are $\omega^{}_{\mathrm{SFF}}$, which controls the onset of random-matrix behavior in the spectral form factor, and the transport frequencies $\omega^{}_{\mathrm{tr}}$, which are derived from transport coefficients. However, a direct quantitative comparison of these frequencies is lacking. Using exact diagonalization, we conduct such a comparison for the spectral functions of current operators in clean and disordered quantum spin ladders with diffusive energy and spin transport. We find clear evidence for the expected low-frequency plateaus in the ETH spectral functions. For the accessible system sizes, $\omega^{}_{\mathrm{SFF}}$ is consistent with the extent of the plateaus, while the transport frequencies are systematically larger and lie in the nonuniversal regime of the spectral functions. Our findings highlight the need to better understand the origin of these quantitative differences.

For a single particle, a quantum information engine can turn measurement fluctuations into transport by raising a barrier behind the particle each time its position is measured. When many particles are present, the barrier still depends on just one number, the position of the leftmost particle, so we let the demon measure that order statistic and nothing else. A demon that resolves every position instead pays a record entropy that grows with particle number, even though the extra distinctions never change where the barrier is placed. We show that the coarser measurement supplies an energy that is bounded independently of particle number, and that this bound yields a ceiling on the record entropy that falls as the filling increases. At the same tilt, the collective engine then delivers $44\%$ more work per recorded nat than independent single-particle engines, each with its own optimized cycle time and each running at equal or greater power per particle. Pauli blocking ends this advantage once the accessible region just above the wall fills. We also evaluate the coherence that the coarse measurement leaves behind, and the cost of placing the barrier imprecisely.

Droplets in complex environments often encounter compositional gradients that drive their propulsion. For example, surfactant gradients induce surface-tension-driven propulsion via the Marangoni effect. Here, we show that such propulsion emerges generically, even in the absence of surfactants. Employing a thin-interface approximation, we derive a compact expression for the droplet velocity in terms of droplet size, viscosity, the sensitivity of surface tension to a regulating component, and the chemical potential gradient of that component. Our theory reveals that droplets move toward regions of lower stability, as encoded in the morphology of the phase diagram, particularly near critical points. Numerical simulations confirm these results, which establish a general route to predicting and designing droplet motility from the phase behavior of complex fluids.

A stochastic optimisation method was recently proposed to efficiently compute large deviation functions, used in statistical physics to characterise the fluctuations of nonequilibrium systems, modelled as Markov processes. The method, called LDSTOP, combines numerical techniques from control theory and machine learning to iteratively construct the "driven process", a controlled version of the Markov process used as a model of nonequilibrium system that realises a given fluctuation or large deviation of that system in an optimal way. Compared to other methods based on spectral approximations, importance sampling, cloning or splitting, LDSTOP is simple, flexible, and scalable, as it works by simulating single trajectories that are gradually guided towards the driven process. Here, we illustrate these advantages by applying the method on the full range of Markov processes considered in applications, namely, discrete-time Markov chains, continuous-time jump processes, and diffusion processes. For each type, we define the objective function to be optimised, explain different options available for representing the driven process (using, e.g., neural networks), and provide implementation details through simple applications. With these contributions, we aim to showcase the method's efficiency, as well as its ease of use when combined with available machine learning packages, such as PyTorch, TensorFlow and JAX, for simulating stochastic processes and solving high-dimensional optimisation problems.

The intermittent, stick-slip flow of amorphous systems is characterized by avalanches whose statistical properties display robust power-laws. In this paper, we study avalanche statistics in a field theoretical model of a soft-glassy system under shear. The order parameter in our model is the fluidity, which is related to the local rate of plastic events in a soft-glass. While avalanche sizes $S$ have been extensively characterized, the statistics of interevent times $t_i$ have remained largely unexplored. Here, we find that both $S$ and $t_i$ are power-law distributed and that their scaling exponents satisfy a precise relation. The model enables the analytical derivation of the scaling exponents and the reproduction of the behavior observed across a wide class of amorphous systems.

Memory is an inherent feature of the dynamics of physical systems. It typically emerges when the complex structure of the system is simplified during the modeling process, obscuring a part of its underlying behavior. Consequently, the correct description of such a physical model requires knowledge of not only its present, but also its past states, leading to non-Markovian dynamics. The resulting interactions depending on the system history, however, introduce a significant layer of mathematical complexity. For this reason, when the memory time is significantly shorter than the time scales characterizing the dynamics, it is typically neglected at the cost of losing information regarding its role in the system behavior. In this dissertation, I introduce a novel methodology, namely the effective mass approach, which bridges the Markovian and non-Markovian regimes and provides an accessible framework for analyzing systems with short memory. In this method, the non-Markovian system is approximated with its Markovian counterpart, in which the short-memory effects are encapsulated within an effective mass. I then utilize this approach to demonstrate that the impact of short memory on physical dynamics can be remarkably pronounced. In particular, I show that the directed transport of a Brownian particle in an environment exhibiting temporal correlations can be reversed relative to its behavior in a memoryless medium. Moreover, I propose a setup wherein memory serves as a~control parameter for the generation of random information bits. Collectively, the results of this dissertation underscore the critical importance of short-time correlations in microscopic systems.

We study the emergence of non-equilibrium steady states (NESS) in stochastic processes under threshold resetting, an event-driven protocol in which the system resets to its initial configuration upon crossing a prescribed spatial boundary (threshold). In contrast to externally driven resetting, whose steady-state properties are well understood, the behavior under threshold resetting remains largely unexplored. We derive general conditions for the existence of a NESS and show that, whenever it exists, the steady state at a given position $x$ can be expressed as the ratio of two fundamental quantities: the mean local time (MLT) at $x$ and the mean first-passage time (MFPT) to hit the threshold. In particular, for noisy systems, a finite MFPT guarantees the existence of a NESS. As an illustrative example, we analyze a drift-diffusion process in one dimension and uncover rich intermediate-time dynamics, including anomalous relaxation in the spatial distribution, damped oscillations in the moments and in the mean-squared displacement (MSD), governed by system parameters. Our findings provide a general understanding of the emergence of non-equilibrium steady states and relaxation dynamics under threshold resetting, revealing how induced events shape the spatial and temporal properties of a broad class of stochastic processes.

We analyse two-dimensional isotropic heterogeneous diffusion with a radially decaying diffusion coefficient $D(\rho)=D_0/\rho^2$, subject to Poissonian stochastic resetting to the origin. Since multiplicative noise makes the process interpretation-dependent, we treat the Itô, Stratonovich, and Hänggi–Klimontovich prescriptions within a single approach. For each, we obtain the exact probability density function, the non-equilibrium stationary state induced by resetting, the mean squared displacement, and the first-passage properties. The non-equilibrium stationary state is governed by modified Bessel functions and departs from the Laplace form found for homogeneous diffusion under resetting, while the mean squared displacement saturates at long times as $\langle\rho^2\rangle\sim\bar{r}^{-1/2}$, $\bar{r}$ being the resetting rate. The mean first-passage time to an absorbing boundary exhibits an optimal resetting rate at which it is minimised. In addition, we incorporate memory effects through a subordination (continuous-time random walk) approach with a power-law kernel, deriving the fractional Fokker–Planck equation together with the non-equilibrium stationary state, the mean squared displacement expressed via the three-parameter Mittag-Leffler function, and the first-passage statistics; the optimal resetting rate increases as the memory exponent decreases.

The exact behaviour of the enveloppes of the single-time and two-time correlators is found for the voter model on a fractal substrate, with nearest-neighbour interactions. Herein the geometry of the fractal substrate is described by its non-integer geometric fractal dimension $d_f$, and its topology and diffusive transport by the distinct spectral dimension $d_s$. On the level of the equations of motion of the correlators this can be modelled by considering a space-dependent diffusion constant ${\cal D}(r)\sim r^{-\theta}$ which implies the spectral index $\theta$, itself a function of $d_f$ and $d_s$. With a scaling ansatz, the generic phenomenology of ageing is confirmed and the dynamic exponent ${z}=2+\theta$ and the autocorrelation exponent $\lambda=d_f$ are derived. The explicitly found dynamic scaling functions are shown to depend only on the spectral dimension $d_s$. The decay of the enveloppe of the density of active interfaces with time is described by the exponent $\alpha=1-d_s/2$ for $d_s<2$, confirming the results of preexisting numerical simulations.

In this work, we characterize the glassy character of the Bose-glass phase in a disordered Bose-Hubbard model using three complementary diagnostics: the finite-temperature spectral function, spatial inhomogeneity, and the inverse participation ratio. Spectral analysis, obtained from finite-temperature Green's function and random phase approximation, shows that disorder introduces localized low-energy states within the Mott gap, eventually closing the gap at sufficiently strong disorder. Spatial inhomogeneity, calculated using the Gutzwiller ansatz, increases sharply with disorder and then saturates at moderate disorder strengths. The inverse participation ratio has been calculated from the exact diagonalization of a small system, and it reveals enhanced localization with stronger disorder. Increasing temperature suppresses these disorder-induced features: the low-energy spectral weight diminishes, the spatial inhomogeneity varies more smoothly, and the IPR decreases. Taken together, these diagnostics show that disorder drives the development of glassy character, while thermal fluctuations gradually wash out its signatures. The early saturation of spatial inhomogeneity compared with the continued evolution of low-energy excitations and the inverse participation ratio demonstrates that no single diagnostic fully captures the evolution of the Bose-glass state, highlighting the need for a combined characterization.

We use the Mori-Zwanzig formalism to derive an exact generalized Langevin equation governing the dynamics of magnetic skyrmions. Using standard approximations, this generalized Langevin equation is found to reduce to the usual Thiele equation. Our approach provides a general route for obtaining extensions of the Thiele equation applicable to magnetic textures and degrees of freedom not covered by the standard derivation route. It also provides some insights into the microscopic justification of the standard Thiele equation, specifically concerning contributions of fast modes to the form of the drift and diffusion terms in the Fokker-Planck equation for skyrmions.