Site Percolation Threshold Research Articles

Site percolation threshold of composite square lattices and its agroecology applications.

We analyze the percolation threshold of square lattices comprising a combination of sites with regular and extended neighborhoods. We found that the percolation threshold of these composed systems smoothly decreases with the fraction of sites with extended neighbors. This behavior can be well-fitted by a Tsallis q-Exponential function. We found a relation between the fitting parameters and the differences in the gyration radius among neighborhoods. We also compared the percolation threshold with the critical susceptibility of nearest and next-to-nearest neighbor monoculture plantations vulnerable to the spread of phytopathogen. Notably, the critical susceptibility in monoculture plantations can be described as a linear combination of two composite systems. These results allow the refinement of mathematical models of phytopathogen propagation in agroecology. In turn, this improvement facilitates the implementation of more efficient computational simulations of agricultural epidemiology that are instrumental in testing and formulating control strategies.

Stereolithographic 3D printing: Formulation design based on percolation thresholds

Stereolithograpy (SLA) is an optical photosolidification technology, which can be used to produce 3D printed dosage forms. In the present work, the model drug 4-Aminosalicylic was combined with the photopolymer poly(ethylene glycol) diacrylate (PEGDA), the plasticizer poly(ethylene glycol) (PEG) and the photoinitiator diphenyl(2,4,6- trimethylbenzoyl) phosphine oxide (DPPO) to form a printable resin. The design of a commercial SLA printer was modified to enable printing of different light-curing mixtures in the same tank. This optimisation allowed to evaluate up to 18 different resins by printing in parallel, thus reducing time and costs. Resulting 3D printed drug delivery systems were analysed with respect to mechanical parameters and drug dissolution rate. The principles of percolation theory were applied, for the first time, to provide a mechanistic insight into the impact of varying PEGDA-PEG ratios. It could be demonstrated that a critical point is reached at 18 %v/v of PEGDA, representing the minimum amount needed in the resins to produce solid objects. This critical point was related to the site percolation threshold of PEGDA. Additional critical points were identified related to the percolation thresholds of PEGDA and PEG in the formulation. We conclude that percolation theory is a valuable concept to obtain mechanistic insights into 3D printed systems manufactured by SLA and to define the Design Space of these innovative dosage forms.

Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone

The site percolation problem is one of the core topics in statistical physics. Evaluation of the percolation threshold, which separates two phases (sometimes described as conducting and insulating), is useful for a range of problems from core condensed matter to interdisciplinary application of statistical physics in epidemiology or other transportation or connectivity problems. In this paper with Newman–Ziff fast Monte Carlo algorithm and finite-size scaling theory the random site percolation thresholds pc for a square lattice with complex neighborhoods containing sites from the sixth coordination zone are computed. Complex neighborhoods are those that contain sites from various coordination zones (which are not necessarily compact). We also present the source codes of the appropriate procedures (written in C) to be replaced in original Newman–Ziff code. Similar to results previously found for the honeycomb lattice, the percolation thresholds for complex neighborhoods on a square lattice follow the power law pc(ζ)∝ζ−γ2 with γ2=0.5454(60), where ζ=∑iziri is the weighted distance of sites in complex neighborhoods (ri and zi are the distance from the central site and the number of sites in the coordination zone i, respectively).

Site percolation in distorted square and simple cubic lattices with flexible number of neighbors.

This paper exhibits a Monte Carlo study on site percolation using the Newmann-Ziff algorithm in distorted square and simple cubic lattices where each site is allowed to be directly linked with any other site if the Euclidean separation between the pair is at most a certain distance d, called the connection threshold. Distorted lattices are formed from regular lattices by a random but controlled dislocation of the sites with the help of a parameter α, called the distortion parameter. The distinctive feature of this study is the relaxation of the restriction of forming bonds with only the nearest neighbors. Owing to this flexibility and the intricate interplay between the two parameters α and d, the site percolation threshold may either increase or decrease with distortion. The dependence of the percolation threshold on the average degree of a site has been explored to show that the obtained results are consistent with those on percolation in regular lattices with an extended neighborhood and continuum percolation.

Percolation and jamming properties in object growth model on a lattice with impurities

Percolation model with nucleation and object growth is studied by Monte Carlo simulations on a triangular lattice with point-like impurities. Growing objects are needle-like objects and self-avoiding random walk chains. In each run through the system the lattice is initially randomly occupied by point-like impurities at given concentration . Then the seeds for the object growth are randomly distributed at given concentration ρ. The percolation properties and the jamming densities are compared for the two classes of growing objects on the basis of the results obtained for a wide range of densities ρ and up to the percolation threshold for the monomer deposition on a triangular lattice. Values of the percolation thresholds have lower values for the needle-like objects than for the self-avoiding random walk chains. The difference is largest for the lowest values of ρ and , and ceases near the values of the site percolation threshold for monomers on the triangular lattice, . Values of the jamming coverage decrease with for given ρ. This effect is more prominent for the growing random walk chains.

Crystal structures, frustrated magnetism, and chemical pressure in Sr-doped Ba3NiSb2O9 perovskites

The effects of chemical pressure on the structural and magnetic properties of the triple perovskite ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ are investigated by substituting ${\mathrm{Sr}}^{2+}$ ions for ${\mathrm{Ba}}^{2+}$ ions. Two ${\mathrm{Ba}}_{3\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ phases could be stabilized via a solid-state reaction at ambient pressure (AP) in air. The $6H$ with ${\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ pairs $(x=0)\ensuremath{\rightarrow}6H$ with $\mathrm{NiSb}{\mathrm{O}}_{9}$ pairs $(x=0.5)\ensuremath{\rightarrow}3C$ (cubic with corner-sharing octahedral, $x=1.25$) sequence of structural phases occurs with increasing Sr content, i.e., chemical pressure, which is like that previously reported for pure samples of ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ obtained under increasing high physical pressure (HP). For the $6H {\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9} (x=0.5)$ phase, using combined Rietveld refinements of powder x-ray and neutron diffraction patterns, precession electron diffraction tomography data collected on thin crystals, aberration-corrected high-angle annular dark field scanning transmission electron microscopy coupled to energy dispersive x-ray spectroscopy mapping, we reach the conclusion that the structure features corner-sharing $\mathrm{Sb}{\mathrm{O}}_{6}$ octahedra and $\mathrm{NiSb}{\mathrm{O}}_{9}$ pairs of face-shared octahedra (or Ni-Sb dumbbells) with either a random orientation of the Ni-Sb dumbbells or nanosized chemical correlations for the dumbbell arrangement. As observed in HP ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ produced through synthesis at 9 GPa, AP ${\mathrm{Ba}}_{1.75}{\mathrm{Sr}}_{1.25}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9} (x=1.25)$ crystallizes in a $3C$ double perovskite ${A}_{2}B{B}^{\ensuremath{'}}{\mathrm{O}}_{6}$ cubic structure where $A, B$, and ${B}^{\ensuremath{'}}$ sites are occupied by (Ba + Sr), Sb, and $(\frac{2}{3}\mathrm{Ni}+\frac{1}{3}\mathrm{Sb})$ atoms, respectively. The ${B}^{\ensuremath{'}}$ sites, which are randomly occupied by spin-1 ${\mathrm{Ni}}^{2+}$ and diamagnetic ${\mathrm{Sb}}^{5+}$, form a face-centered-cubic (FCC) sublattice where the ${\mathrm{Ni}}^{2+}$ amount stays above the site percolation threshold. Weiss temperatures ($\ensuremath{\approx}\ensuremath{-}65$ and $\ensuremath{\approx}\ensuremath{-}213$ K for ${\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ and ${\mathrm{Ba}}_{1.75}{\mathrm{Sr}}_{1.25}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$, respectively) indicate that dominant magnetic interactions between ${\mathrm{Ni}}^{2+}$ spins are antiferromagnetic with magnitudes like those observed in the corresponding HP phases of pure ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$. As for the $6H$ HP ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ compound, in $6H {\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$, muon spin relaxation $(\ensuremath{\mu}\mathrm{SR})$ measurements identify a dynamic magnetic state down to the base temperature (95 mK), consistent with a previously published inelastic neutron scattering study. For $3C {\mathrm{Ba}}_{1.75}{\mathrm{Sr}}_{1.25}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}, \ensuremath{\mu}\mathrm{SR}$ and $^{121}\mathrm{Sb}$ nuclear magnetic resonance measurements both indicate the presence of a transition to a static magnetic state below $11(1)$ K with a significant amount of disorder in this frozen state, in contrast to the spin-liquid state previously suggested for the $3C$ HP phase of ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$. Consistently, a broad maximum is observed in the specific heat at the same temperature. Building on the structural data, the magnetic properties of HP $6H {\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ and AP $6H {\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ are discussed in light of recent works on triangular and ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ honeycomb systems with or without quenched disorder. We are led to the conclusion that the driving force toward a spin-liquid-like state is quenched disorder which needs to be incorporated in ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ honeycomb models. Our evidence of a magnetic transition to a frozen magnetic ground state for the AP Sr-doped $3C$ phase is in line with models for geometrically frustrated FCC antiferromagnets. This calls for a better experimental and possibly theoretical understanding of the HP $3C$ phase.

Random site percolation on honeycomb lattices with complex neighborhoods.

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3-24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number z of sites in the neighborhoods, the site percolation thresholds p follow the dependency p ∝ 1 / z, as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of p corresponding to the same number z of sites in the neighborhoods). An example of a single-value index ζ = ∑ i z r-where z and r are the number of sites and radius of the ith coordination zone, respectively-characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence p ∝ 1 / ζ. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.

New bounds for the site percolation threshold of the hexagonal lattice

The site percolation threshold of the hexagonal lattice satisfies 0.656 246 < p c < 0.739 695. For comparison, the largest previous lower bound of 0.652 703… was established in 1981, and the smallest previous upper bound of 0.743 359 was derived in 2007. The bound is obtained by using the substitution method to compare the hexagonal lattice site model to an exactly-solved two-parameter site percolation model on the martini lattice. Computational reductions involving graph-welding, symmetry, non-crossing partitions, and network flow computations overcome challenges to establishing stochastic ordering between the models.

On the estimation of percolation thresholds for real networks

The percolation threshold is the critical point at which a giant connected component emerges for the first time in the percolation process. Its theoretical estimation has recently gathered much attention, especially for real networks with complicated structures. However, the theoretical methods so far mainly provide lower bounds of the percolation threshold with non-negligible errors. In this paper, we first generalize the existing message passing algorithm by considering the independence of variables in iteration equations. We then give the exact implicit expressions of the site and bond percolation thresholds based on the correction δs(b). The experimental results show that the correction is strongly related to network structural properties so that we can predict it by the gradient boosting regression model. Finally, we take the predicted correction δs(b) to our theoretical framework and show that our estimates of site and bond percolation thresholds on 209 real networks are more accurate than those of the state-of-the-art methods.

Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions.

Extended-range percolation on various regular lattices, including all 11 Archimedean lattices in two dimensions and the simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number z and site thresholds p_{c} for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting z versus 1/p_{c} and z versus -1/ln(1-p_{c}) using the data of d'Iribarne etal. [J. Phys. A 32, 2611 (1999)JPHAC50305-447010.1088/0305-4470/32/14/002] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of zp_{c}∼4η_{c}=4.51235, where η_{c} is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for bcc and fcc lattices with 2nd and 3rd NN, and bond thresholds for the sc lattice with up to the 13th NN, are obtained by Monte Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of zp_{c}∼8η_{c}=2.7351, where η_{c} is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior p_{c}=1/(z-1) is expected to hold for large z, and the finite-z correction is confirmed to satisfy zp_{c}-1∼a_{1}z^{-x}, with x=2/3 for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21, 56 (2016)1083-648910.1214/16-EJP6] and by Xu etal. [Phys. Rev. E 103, 022127 (2021)2470-004510.1103/PhysRevE.103.022127]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of zp_{c} has universal properties, depending only on the dimension of the system and whether site or bond percolation but not on the type of lattice.

Percolation thresholds on high-dimensional D_{n} and E_{8}-related lattices.

The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now also facilitates the study of D_{n} root lattices in n dimensions as well as E_{8}-related lattices. Here, we consider the percolation problem on D_{n} for n=3 to 13 and on E_{8} relatives for n=6 to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion based on lattice animal enumeration for D_{n} lattices. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as n increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends. Interestingly, the finite-size scaling exponent for invasion percolation is found to be lattice and percolation-type specific.

Simple peeling of planar maps with application to site percolation

AbstractThe peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolations on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations, distinct vertices on the exploration contour may, at latter stage, be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative “simple” peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way, we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the “dense regime,” where the half-planar random Boltzmann map does not possess an infinite simple core.

Empirical formula for site and bond percolation thresholds on Archimedean and 2-uniform lattices

The problem of percolation on Archimedean and 2-uniform lattices is investigated. An empirical formula is postulated for the ratio between the site percolation threshold pcs, and the bond percolation threshold pcb. The expression is pcs∕pcb≈az̄s3∕z̄s−12, where a=0.16428 and z̄s is the mean number of nearest neighbors of a given lattice site. For each of the 11 Archimedean and 20 2-uniform lattices which constitute our sample, a good correspondence is obtained with previous calculations of pcs∕pcb. The formula is found to be also valid for standard percolation in three-dimensional lattices and for percolation of tortuous k-mers [objects occupying k adjacent lattice elements (sites or bonds)] on square lattices.

Distribution of the random walk conditioned on survival among quenched Bernoulli obstacles

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d \geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $d\log_{1/p}n$. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest.

Site percolation thresholds on triangular lattice with complex neighborhoods.

We determine thresholds pc for random site percolation on a triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN), and next-next-next-next-nearest (5NN) neighbors, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, and 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [Phys. Rev. E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [Phys. Rev. E 90, 062101 (2014)], for estimating thresholds from low statistics data. The estimated values of percolation thresholds are pc(4NN)=0.192410(43), pc(3NN+2NN)=0.232008(38), pc(5NN+4NN)=0.140286(5), pc(3NN+2NN+NN)=0.215484(19), pc(5NN+4NN+NN)=0.131792(58), pc(5NN+4NN+3NN+2NN)=0.117579(41), and pc(5NN+4NN+3NN+2NN+NN)=0.115847(21). The method is tested on the standard case of site percolation on the triangular lattice, where pc(NN)=pc(2NN)=pc(3NN)=pc(5NN)=12 is recovered with five digits accuracy pc(NN)=0.500029(46) by averaging over one thousand lattice realizations only.

Voronoi chains, blocks, and clusters in perturbed square lattices

Perturbed lattices provide simple models for studying many physical systems. In this paper we study the distribution of Voronoi chains, blocks, and clusters with prescribed combinatorial features in the perturbed square lattice, generalizing earlier work. In particular, we obtain analytic results for the presence of hexagonally-ordered regions within a square-ordered phase. Connections to high-temperature crystals and to non-equilibrium phase transitions are considered. In an appendix, we briefly consider the site-percolation threshold for this system.

Site percolation and isoperimetric inequalities for plane graphs

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.

Two transitions in spatial modular networks

Understanding the resilience of infrastructures, such as a transportation network, has significant importance for our daily life. Recently, a homogeneous spatial network model was developed for studying spatial embedded networks with characteristic link length such as power-grids and the brain. However, although many real-world networks are spatially embedded and their links have characteristics length such as pipelines, power lines or ground transportation lines they are not homogeneous but rather heterogeneous. For example, density of links within cities are significantly higher than between cities. Here we develop and study numerically and analytically a similar realistic heterogeneous spatial modular model using percolation process to better understand the effect of heterogeneity on such networks. The model assumes that inside a city there are many lines connecting different locations, while long lines between the cities are sparse and usually directly connecting only a few nearest neighbours cities in a two dimensional plane. We find that our heterogeneous model experiences two distinct continuous transitions, one when the cities disconnect from each other and the second when each city breaks apart. This is in contrast to the homogeneous model where a single transition is found. Although the critical threshold for site percolation in 2D grid remains an open question we analytically find the critical threshold for site percolation in this model. In addition, it has been found that the homogeneous model experience a single transition having a unique phenomenon called critical stretching where a geometric crossover from random to spatial structure in different scales found to stretch non-linearly with the characteristic length at criticality. In marked contrast, we show here that the heterogeneous model does not experience such a phenomenon indicating that critical stretching strongly depends on the network structure.

Borealis—A generalized global update algorithm for Boolean optimization problems

Optimization problems with Boolean variables that fall into the nondeterministic polynomial (NP) class when cast as decision problems are of fundamental importance in computer science, mathematics, physics and industrial applications. Most notably, solving constraint-satisfaction problems, which are related to spin-glass-like Hamiltonians in physics, remains a difficult numerical task. As such, there has been great interest in designing efficient heuristics to solve these computationally difficult problems. Inspired by parallel tempering Monte Carlo in conjunction with the rejection-free isoenergetic cluster algorithm developed for Ising spin glasses, we present a generalized global update optimization heuristic that can be applied to different NP-complete problems with Boolean variables. The global cluster updates allow for a wide-spread sampling of phase space, thus considerably speeding up optimization. By carefully tuning the pseudo-temperature (needed to randomize the configurations) of the problem, we show that the method can efficiently tackle optimization problems with over-constraints or on topologies with a large site percolation threshold. We illustrate the efficiency of the heuristic on paradigmatic optimization problems, such as the maximum satisfiability problem and the vertex cover problem. Because this physics-based algorithm is an algorithm that searches solutions globally, it performs best for random Max-k-SAT instances.

Localization for Random Walks Among Random Obstacles in a Single Euclidean Ball

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that for environments with probability tending to one as $n\to \infty$ there exists a unique discrete Euclidean ball of volume $d \log_{1/p} n$ asymptotically such that the following holds: conditioned on survival up to time $n$ we have that at any time $t \in [o(n),n]$ with probability tending to one the simple random walk is in this ball. This work relies on and substantially improves a previous result of the authors on localization in a region of volume poly-logarithmic in $n$ for the same problem.