We study a multiple invasion model to simulate corrosion or intrusion processes. Estimated values for the fractal dimension of the invaded region reveal that the critical exponents vary as function of the generation number $G$, i.e., with the number of times the invasion process takes place. The averaged mass $M$ of the invaded region decreases with a power-law as a function of $G$, $M\sim G^β$, where the exponent $β\approx 0.6$. We also find that the fractal dimension of the invaded cluster changes from $d_{1}=1.887\pm0.002$ to $d_{s}=1.217\pm0.005$. This result confirms that the multiple invasion process follows a continuous transition from one universality class (NTIP) to another (optimal path). In addition, we report extensive numerical simulations that indicate that the mass distribution of avalanches $P(S,L)$ has a power-law behavior and we find that the exponent $τ$ governing the power-law $P(S,L)\sim S^{-τ}$ changes continuously as a function of the parameter $G$. We propose a scaling law for the mass distribution of avalanches for different number of generations $G$.
The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000^2 and 120^3 respectively. A soft spin O(2) model, in the same universality class as the 3D XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K_c=0.45412(2) for the 3D XY model and eta=0.037(2) for the 3D XY universality class. For the 2D XY model the results are K_c=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the 2D or 3D XY models.
We present a detailed study of the Equilibriumlike invaded cluster algorithm (EIC), recently proposed as an extension of the invaded cluster (IC) algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is discussed in context of dynamical properties. We also discuss the relation to the Li-Sokal bound for the dynamical exponent $z$.
The invaded cluster approach is extended to 2D Potts model with annealed vacancies by using the random-cluster representation. Geometrical arguments are used to propose the algorithm which converges to the tricritical point in the two-dimensional parameter space spanned by temperature and the chemical potential of vacancies. The tricritical point is identified as a simultaneous onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of "geometrical disorder cluster". The location of the tricritical point and the concentration of vacancies for q = 1, 2, 3 are found to be in good agreement with the best known results. Scaling properties of the percolating scaling cluster and related critical exponents are also presented.
The Invaded Cluster (IC) dynamics introduced by Machta et al. [Phys. Rev. Lett. 75 2792 (1995)] is extended to the fully frustrated Ising model on a square lattice. The properties of the dynamics which exhibits numerical evidence of self-organized criticality are studied. The fluctuations in the IC dynamics are shown to be intrinsic of the algorithm and the fluctuation-dissipation theorem is no more valid. The relaxation time is found very short and does not present critical size dependence.
We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric Tübingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But also notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and
We propose an extension of the nonequilibrium invaded cluster (IC) algorithm, which reestablishes a correct scaling of fluctuations at criticality and also self-adjusts to the critical temperature. We show that by introducing a single constraint to the intrinsic quantity of the IC algorithm the temperature becomes well defined and the sampling of the equilibrium ensemble is regained. The procedure is applied to the Potts model in two and three dimensions.
A parallel version of the invaded cluster algorithm is described. Results from large scale (up to 4096^2 and 512^3) simulations of the Ising model are reported. No evidence of critical slowing down is found for the three-dimensional Ising model. The magnetic exponent is estimated to be 2.482 \pm .001 (beta/nu = 0.518 pm .001) for the three-dimensional Ising model.
A new cluster algorithm based on invasion percolation is described. The algorithm samples the critical point of a spin system without a priori knowledge of the critical temperature and provides an efficient way to determine the critical temperature and other observables in the critical region. The method is illustrated for the two- and three-dimensional Ising models. The algorithm equilibrates spin configurations much faster than the closely related Swendsen-Wang algorithm.
The invaded cluster algorithm, a new method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three dimensions. The algorithm is shown to be useful for both first-order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities. The dynamic properties of the invaded cluster algorithm are studied. Numerical evidence suggests that the algorithm has no critical slowing for Ising models.
Simulations of the two-dimensional Ising and 3-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sub-lattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time tau(l,L) for an l*l sub-lattice attains a maximum in the crossover region, and a dynamic exponent z for the IC algorithm is defined according to tau_max ~ L^z. Simulation results for the 3-state model yield z=.346(.002) which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that z<.21 and possibly that tau_max ~ log L so that z=0 -- similar to previous results for the SW and Wolff algorithms.
A superpotential deformation that is cubic in one of the chiral superfields of ABJM makes the latter theory flow into a new, ${\cal N}=2$ superconformal phase. This is holographically dual to a warped $\textrm{AdS}_4 \times_w S^7$ solution of M-theory equipped with a squashed and stretched metric on $S^7$. We determine the spectrum of spin-2 operators of the cubic deformation at low energies by computing the spectrum of Kaluza-Klein (KK) gravitons over the dual AdS$_4$ solution. We calculate, numerically, the complete graviton spectrum and, analytically, the spectrum of gravitons that belong to short multiplets. We also use group theory to assess the structure of the full KK spectrum, and conclude that ${\cal N}=2$ supermultiplets cannot be allocated KK level by KK level. This phenomenon, usually referred to as "space invaders scenario", is also known to occur for another $\textrm{AdS}_4$ solution based on a different squashed $S^7$.
The Fisher-KPP equation is the canonical reaction-diffusion equation used in the study of invasive phenomena in mathematical biology. While moving boundaries have already been included to introduce biologically realistic sharp interfaces, these flux-based conditions assume loss of mass as a byproduct of boundary movement. We consider a mass-conserving boundary condition at a free boundary of the Fisher-KPP equation to allow interface movement without population consumption. We use a combination of phase plane analysis, perturbation analysis, and numerical simulation to show that the model supports both invading and receding travelling wave solutions with distinct wave speeds and front densities. Prescribing the free boundary velocity as a linear function of the front density, we show the existence of multiple stable and unstable travelling wave solutions, as well as receding population blow-up. Our results indicate that in regions where multiple steady states exist, those with larger wave speeds are stable. This is corroborated by considering the parameter regime associated with the uniform steady state solution and performing linear stability analysis. The framework we develop is
It is well known that a continuous phase transition in Bernoulli bond percolation on the integer lattice is equivalent to a vanishing probability a vertex is invaded in invasion percolation. We provide a coupling between invasion percolation and first passage percolation with log-uniform passage times. This yields a new equivalent condition for a continuous phase transition in bond percolation.
Using a reaction-diffusion model with free boundaries in one space dimension for a single population species with density $u(t,x)$ and population range $[g(t), h(t)]$, we demonstrate that the Allee effects can be eliminated if the species maintains its population density at a suitable level at the range boundary by advancing or retreating the fronts. It is proved that with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density level at the range boundary is maintained at other levels. If the invading cane toads in Australia used this strategy at the range boundary to become a super invader, then our results may explain why toads near the invading front evolve to have longer legs and run faster.
In this paper, we study a special case of the invasion fitness matrix in a replicator equation: the invader-driven case. In this replicator, each species is defined by its unique active invasiveness potential (initial growth rate when rare), upon invading any other species, independently of the partner. We derive explicit expressions and theorems to fully characterize the steady-states of this system, including its unique interior coexistence regime, reached for positive species traits, or alternative boundary exclusion states, reached for negative species traits. We study the internal stability of coexistence steady-states, and the system's stability to outsider invasion, relevant for system assembly. We provide detailed analytical results for critical diversity thresholds, and for the special case of random uniform species traits, we analytically compute the probability of stable $k-$species coexistence in a random pool of size $N$, and show that the mean number of co-existing species can be approximated as $\mathbb{E}[n] \sim \sqrt{2N}$. We also derive explicit mathematical conditions for invader traits and invasion outcomes (augmentation, rejection, and replacement), dependent
Despite the remarkable success, recent reconstruction-based anomaly detection (AD) methods via diffusion modeling still involve fine-grained noise-strength tuning and computationally expensive multi-step denoising, leading to a fundamental tension between fidelity and efficiency. In this paper, we propose InvAD, a novel inversion-based anomaly detection approach ("detection via noising in latent space") that circumvents explicit reconstruction. Importantly, we contend that the limitations in prior reconstruction-based methods originate from the prevailing "detection via denoising in RGB space" paradigm. To address this, we model AD under a reconstruction-free formulation, which directly infers the final latent variable corresponding to the input image via DDIM inversion, and then measures the deviation based on the known prior distribution for anomaly scoring. Specifically, in approximating the original probability flow ODE using the Euler method, we enforce only a few inversion steps to noise the clean image to pursue inference efficiency. As the added noise is adaptively derived with the learned diffusion model, the original features for the clean testing image can still be lever
Evolutionary approaches have previously been shown to be effective learning methods for a diverse set of domains. However, the domain of game-playing poses a particular challenge for evolutionary methods due to the inherently large state space of video games. As the size of the input state expands, the size of the policy must also increase in order to effectively learn the temporal patterns in the game space. Consequently, a larger policy must contain more trainable parameters, exponentially increasing the size of the search space. Any increase in search space is highly problematic for evolutionary methods, as increasing the number of trainable parameters is inversely correlated with convergence speed. To reduce the size of the input space while maintaining a meaningful representation of the original space, we introduce Sparse Cosine Optimized Policy Evolution (SCOPE). SCOPE utilizes the Discrete Cosine Transform (DCT) as a pseudo attention mechanism, transforming an input state into a coefficient matrix. By truncating and applying sparsification to this matrix, we reduce the dimensionality of the input space while retaining the highest energy features of the original input. We dem
In this work, we investigate the long-time dynamics of a two species competition model of Lotka-Volterra type with nonlocal diffusions. One of the species, with density $v(t,x)$, is assumed to be a native in the environment (represented by the real line $\R$), while the other species, with density $u(t,x)$, is an invading species which invades the territory of $v$ with two fronts, $x=g(t)$ on the left and $x=h(t)$ on the right. So the population range of $u$ is the evolving interval $[g(t), h(t)]$ and the reaction-diffusion equation for $u$ has two free boundaries, with $g(t)$ decreasing in $t$ and $h(t)$ increasing in $t$, and the limits $h_\infty:=h(\infty)\leq \infty$ and $g_\infty:=g(\infty)\geq -\infty$ thus always exist. We obtain detailed descriptions of the long-time dynamics of the model according to whether $h_\infty-g_\infty$ is $\infty$ or finite. In the latter case, we reveal in what sense the invader $u$ vanishes in the long run and $v$ survives the invasion, while in the former case, we obtain a rather satisfactory description of the long-time asymptotic limit for both $u(t,x)$ and $v(t,x)$ when a certain parameter $k$ in the model is less than 1. This research is co