Large Deviations of Convex Hulls of Self-Avoiding Random Walks
A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely the classical Self-Avoiding Walk, the Smart-Kinetic Self-Avoiding Walk, and the Loop-Erased Random Walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of the support down to probabilities far smaller than . We give an approximate closed form of the so-called large-deviation rate function which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further we show correlations between the two observables also in the limits of atypical large or small values.
pacs:02.50.-r, 75.40.Mg, 89.75.Da
The standard random walk is a simple Markovian process, which has a history as a model for diffusion. There are many exact results known Hughes (1996). If memory is added to the model, e.g., to interact with the past trajectory of the walk, analytic treatment becomes much harder. A class of self-interacting random walks that we will focus on in this study, are self-avoiding random walks, which live on a lattice and do not visit any site twice. This can be used to model systems with excluded volume, e.g., polymers whose single monomers can not occupy the same site at once Madras and Slade (2013). There are more applications which are not as obvious, e.g., a slight modification of the Smart-Kinetic Self-Avoiding Walk traces the perimeter of critical percolation clusters Weinrib and Trugman (1985), while the Loop-Erased Random Walk can be used to study spanning trees Manna et al. (1992) (and vice versa Majumdar (1992)).
One of the central properties of random walk models is the exponent , which characterizes the growth of the end-to-end distance with the number of steps , i.e., . While this has the value for the standard random walk, its value is larger for the self-avoiding variations, which are effectively pushed away from their past trajectory. In two dimensions, this value (and other properties) can often be obtained by the correspondence to Schramm-Loewner evolution Cardy (2005); Lawler et al. (2002, 2011); Kennedy (2015). But between two dimensions and the upper critical dimension, above which the behavior is the same as the standard random walk, Monte Carlo simulations are used to obtain estimates for the exponent .
Here we want to study the convex hulls of a selection of self-avoiding walk models featuring larger values of . The convex hull allows one to obtain a global picture of the space occupied by a walk, without exposing all details of the walk. As an example, convex hulls are used to describe the home ranges of animals Mohr (1947); Worton (1987); Boyle et al. (2009). Namely, we will look at the Smart-Kinetic Self-Avoiding Walk (SKSAW), the classical Self-Avoiding Walk (SAW) and the Loop-Erased Random Walk (LERW), since they span a large range of values and are well established in the literature. About the convex hulls of standard random walks we already know plenty properties. The mean perimeter and area are known exactly since over 20 years Gérard Letac (1980); Letac (1993) for large walk lengths , i.e., the Brownian Motion limit. Since then simpler and more general methods were devised, which are based on using Cauchy’s formula with relates the support function of a curve to the perimeter and the area enclosed by the curve Randon-Furling et al. (2009); Majumdar et al. (2010). More recently also the mean hypervolume and surface for arbitrary dimensions was calculated Eldan (2014). For discrete-time random walks with jumps from an arbitrary distribution, the perimeter of the convex hull for finite (but large) walk lengths were computed explicitly Grebenkov et al. (2017). For the case of Gaussian jump lengths even an exact combinatorial formula for the volume in arbitrary dimensions is known Kabluchko and Zaporozhets (2016). For the variance there is an exact result for Brownian bridges Goldman (1996). Concerning the full distributions, no exact analytical results are available. Here sophisticated large-deviation simulations were used to numerically explore a large part of the full distribution, i.e., down to probabilities far smaller than Claussen et al. (2015); Dewenter et al. (2016); Schawe et al. (2017).
Despite this increasing interest in the convex hulls of standard random walks, there seem to be no studies treating the convex hulls of self-avoiding walks. To fill this void, we use Markov chain Monte Carlo sampling to obtain the distributions of some quantities of interest over their whole support. To connect to previous studies Claussen et al. (2015); Dewenter et al. (2016); Schawe et al. (2017) we also compare the aforementioned variants to the standard random walk on a square lattice (LRW). We are mainly interested in the full distribution of the area and the perimeter of dimensional hulls for walks in the plane, since the effects of the self-interactions are stronger in lower dimensions. Though, we will also look into the volume in the dimensional case. In the past study on standard random walks Schawe et al. (2017) we found that the full distribution can be scaled to a universal distribution using only the exponent and the dimension for large walk lengths . For the present case, where a walk might depend on its full history, one could expect a more complex behavior. Nevertheless, our results presented below show convincingly that also for self-interacting walks the distributions are universal and governed mainly by the exponent , except for some finite-size effects, which are probably caused by the lattice structure. Further we use the distributions to obtain empirical large-deviation rate functions Touchette (2009), which suggests that a limiting rate function is mathematically well defined. We also give an estimate for the rate function, which is compatible with the known case of standard random walks and with all cases under scrutiny in this study.
Ii Models and Methods
This sections gives a short overview over the used models and methods, with references to literature more specialized on the corresponding subject. Where we deem adequate, also technical details applicable for this study are mentioned.
ii.1 Sampling Scheme
To generate the whole distribution of the area or perimeter of the convex hull of a random walk over its full support, a sophisticated Markov chain Monte Carlo (MCMC) sampling scheme is applied Hartmann (2002, 2011). The Markov chain is here a sequence of different walk configurations. The fundamental idea is to treat the observable , i.e., the perimeter, area or volume, as the energy of a physical system which is coupled to a heat bath with adjustable “temperature” and to sample its equilibrium distribution using the Markov chain. This can be easily done using the classical Metropolis algorithm Metropolis et al. (1953). Therefore the current walk configuration is changed a bit (the precise type of change is dependent on the type of walk, we are looking at and is explained in the following sections). The change is accepted with the acceptance probability
and rejected otherwise. The will then bias the configuration towards specific ranges of . Configurations at small and negative will show larger than typical , small and positive show smaller than typical and large values independent of the sign sample configurations from the peak of the distribution. Fig. 1 shows typical walk configurations of the self-avoiding walk at different values of .
In a second step, histograms of the equilibrium distribution are corrected for the bias introduced via as follows.
The free parameter can be obtained by enforcing continuity and normalization of the distribution. We do not present further details here, because the algorithm Hartmann (2002) has been applied and explained in detail several times, also in a very general form Hartmann (2014). In particular, the algorithm was already used successfully in other studies looking at the large deviation properties of convex hulls of random walks Claussen et al. (2015); Dewenter et al. (2016).
ii.2 Lattice Random Walk (LRW)
All of the self-interacting random walks, which are the focus of this study, are typically treated on a lattice. Hence, we will start by introducing the simple, i.e., non-interacting, isotropic random walk on a lattice. For simplicity we will use a square lattice with a lattice constant of . A realization consists of randomly chosen discrete steps . Here we use steps between adjacent lattice sites, i.e., -dimensional Cartesian base vectors , which are drawn uniformly from . The realization can be defined as the tuple of the steps and the position at time as
Here we set the start point at the coordinate origin. The set of visited sites is therefore
The central quantity of the LRW is the average end-to-end distance
where denotes the average over the disorder. It grows polynomially and is characterized by the exponent via . For the LRW it is , which is typical for all diffusive processes.
As the change move for the Metropolis algorithm, we replace a randomly chosen by a new randomly drawn displacement. Since our quantity of interest is the convex hull, i.e., a global property of the walk, we do not profit much from local moves, e.g., crankshaft moves. Thus we use this simple, global move.
ii.3 Smart-Kinetic Self-Avoding Walk (SKSAW)
The Smart-Kinetic Self-Avoiding Walk (SKSAW) Weinrib and Trugman (1985); Kremer and Lyklema (1985) is probably the most naive approach to a self-avoiding walk. It grows on a lattice and never enters sites it already visited. Since it is possible to get trapped on an island inside already visited sites, this walk needs to be smart enough to never enter such traps.
In it is possible to avoid traps using just local information in constant time using the winding angle method Kremer and Lyklema (1985). In conjunction with hash table backed detection of occupied sites, a realization with steps can be constructed in time .
This method will typically yield longer stretched walks than the LRW, due to the constraint that it needs to be self-avoiding. This can be characterized by the exponent , which is larger than in .
The sketch Fig. 3 shows that this ensemble does not contain every configuration with the same probability but prefers closely winded configurations. This is also visible in Fig. 2(b). This is characterized by the exponent Kennedy (2015) which is larger than the for LRW, but smaller than for the SAW. Also note that it is conjectured that the upper critical dimension is Kremer and Lyklema (1985), i.e., for all – possibly with logarithmic corrections in . Therefore only is simulated in this study.
While it is easy to draw realizations from this ensemble uniformly, i.e., simple sampling, it is not so straight forward to apply the MCMC changes. If one just changes single steps like for the LRW, and accepts if it is self-avoiding or rejects if it is not, one will generate all self-avoiding walk configurations with equal probability. Our approach to generate realizations according to this ensemble handles the construction of the walk as a black box. It acts on the random numbers used to generate a realization from scratch. During the MCMC at each iteration one random number is replaced by a new random number and a SKSAW realization is regenerated from scratch using the modified random numbers Hartmann (2014). This change is then accepted according to Eq. (1) and undone otherwise.
ii.4 Self-Avoiding Random Walk (SAW)
While the above mentioned SKSAW does produce self-avoiding walks, SAW denotes another ensemble. The ensemble where realizations are drawn uniformly from the set of all self-avoiding configurations. It is not trivial to sample from this distribution efficiently. The black box method used for SKSAW is not feasible, since the construction of a SAW takes time exponential in the length with simple methods like dimerization Suzuki (1968); Madras and Slade (2013). It is possible to perform changes directly on the walk configuration and accept them according to Eq. (1), but their rejection rate is typically quite high and the resulting configurations are very similar Madras and Slade (2013), which makes this inefficient. The state of the art method to sample SAW is the pivot algorithm Madras and Slade (2013). It chooses a random point and uses it as the pivot for a random symmetry operation, i.e., rotation or mirroring. If the resulting configuration is not self avoiding, it is rejected. Otherwise we accept it with the temperature dependent acceptance probability Eq. (1).
As mentioned previously, the exponent Lawler et al. (2002) is larger than for the SKSAW. Since the upper critical dimension for SAW is , this study will also look at , where an exact value of is not known and the best estimate is Clisby (2010a), though our focus is on for this type.
ii.5 Loop-Erased Random Walk (LERW)
The LERW Lawler (1980) uses a different approach to achieve the self-avoiding property. It is built as a simple LRW but each time a site is entered for the second time, the loop that is formed, i.e., all steps since the first entering of this site, is erased. While this ensures no crossings in the walk, the resulting ensemble is different from the SAW ensemble and the walks are longer stretched out, as characterized by the larger exponent Lawler et al. (2011); Guttmann and Bursill (1990); Majumdar (1992). Similar to the SAW the upper critical dimension is and an estimate for is Wilson (2010).
For construction – similar to SKSAW – we need to keep all used random numbers and change them in the MCMC algorithm. This leads to a dramatically higher memory consumption than simple sampling, where each loop can be discarded as soon as it is closed.
ii.6 Convex Hulls
We will study the convex hulls of the sites visited by the random walk . The convex hull of a point set is the smallest polytope containing all Points and all line segments . Some example hulls are shown in Fig. 2.
Convex hulls are one of the most basic concepts in computational geometry 1113 of the first 4 examples for static problems of computational geometry in the Wikipedia can utilize convex hulls for their solution (https://en.wikipedia.org/wiki/Computational_geometry, 12.01.2018). with noteworthy application in the construction of Voronoi diagrams and Delaunay triangulations Brown (1979).
For point sets in the plane, we use Andrew’s Monotone Chain Andrew (1979) algorithm for its simplicity and Quickhull Bykat (1978) as implemented by qhull Barber et al. (1996) for . Both algorithms have a time complexity of . In Andrew’s Monotone Chain algorithm results in ordered points of the convex hull. Adjacent points in this ordering are the line segments of the convex hull. Quickhull results in the simplical facets of the convex hull.
To obtain the perimeter of a convex hull, we sum the lengths of its line segments . To calculate the area and the volume, we use the same fundamental idea. In both cases we subdivide the area/volume into simplexes, i.e., triangles for the area and tetrahedra for the volume. Therefore we choose an arbitrary fixed point inside of the convex hull and construct a simplex for each facet , i.e., for each line segment of the hull forms a triangle and each triangular face of a dimensional polyhedron, forms a tetrahedron with . The volume of a triangle is trivially
where is the perpendicular distance from a facet to a point . Since the union of all triangles built this way, is the whole polygon, the sum of their areas is the area of the polygon. Similar the volume of a polyhedron is the sum of the volumes of all tetrahedra constructed from its faces. The volume of the individual tetrahedra is given by
For random walks on a lattice with steps of length in dimensions the maximum volume is
for divisible by the effective dimension of the observable, e.g., 2 for the area of a planar hull or 3 for the volume in three dimensions. For example, the configuration of maximum area corresponds to an L-shape, i.e., . This form can be derived by the general volume of an -dimensional simplex defined by its vertices Stein (1966)
Without loss of generality, we set to be the coordinate origin. To achieve maximum volume all need to be orthogonal and of equal length. Thus a random walk going steps along some base vector and continuing with steps in direction has a convex hull defined by the tetrahedron specified by . The matrix is thus triangular and its determinant is the product of its diagonal entries which leads directly to Eq. (5). An exception occurs in where the perimeter is .
The focus of this work lies on dimensional SAW and LERW. The results for higher dimensions and for SKSAW are generated with less numerical accuracy. The LRW results also have a lower accuracy as their purpose is mainly to scrutinize the effect of the lattice structure underlying all considered walk types in comparison to the non-lattice results from Schawe et al. (2017). Also not all combinations are simulated, but only those listed with a value in Table 2.
The same raw data is evaluated for equidistant bins and logarithmic bins. And the respective variants are shown according to the scaling of the -axis.
To get an intuition for how the configurations with atypical large areas or perimeters look like, we visualize the correlation between these two observables as scatter plots in Fig. 4.
Since the smallest possible SAW is an (almost) fully filled square, there can not be instances below some threshold, which explains the gaps on the left side of the scatter plots and of the distributions shown in the following section. In the center of the scatter plots, which is already in probability regions far beyond the capabilities of simple sampling methods, the behavior becomes strongly dependent on the bias.
If biasing for large perimeters (top) the area shows a non-monotonous behavior. First, somehow larger perimeters come along typically with larger areas for entropic reasons, i.e., there are less configurations which are long and thin, and more bulky, which have a larger area. Though, for the far right tail, the only configurations with extreme large perimeters are almost line like and have thus a very small area. Also note that the excluded volume effect of the SAW leads to overall larger areas at the same perimeters.
On the other hand, when biasing for large areas (bottom) the configurations with largest area, which are L-shaped (cf. Fig. 1), unavoidably have quite large perimeters, hence the scatter plots show an almost linear correlation between area and perimeter. Since the configurations of large areas naturally avoid self intersections, since steps on already visited points do not enlarge the convex hull, the differences between LRW and SAW diminish in the right tail. Note that with the large-area bias, no walks with the very extreme perimeters exist, for the reason already mentioned.
Note however that these scatter plots are very dependent on which observable we are biasing for. In principle we observe that small perimeters are strongly correlated with small areas while for large but not too large perimeters, there is a broad range of area sizes possible. For extremely large perimeters, the area must be small. For a comprehensive analysis, one would need a full two dimensional histogram, wich could be obtained using Wang Landau sampling, but which is beyond the scope of this study and would require a much larger numerical effort. Nevertheless, from looking at Fig. 4 one can anticipate that the two dimensional histogram would exhibit a strong correlation for small values of and a broad scatter of the accessible values of for larger but not too large values of .
iii.2 Moments and Distributions
The distributions of the different walk types differ considerably. This can be observed in Fig. 5, where distributions of the area for all types with steps are drawn. The main part of the distribution shifts to larger values for larger value of as expected and the probability of atypically large areas is boosted even more in the tails.
In the right tail, the distributions seem to bend down. Below, where we show results for different walk sizes , we see that this is a finite-size effect of the lattice structure and the fixed step length. This can be seen also as follows: Since the lattice together with the fixed step length sets an upper bound on the area, the probability plummets near this bound for entropic reasons, i.e., there are for any walk length only 8 configurations with maximum area (due to symmetries) such that all self-avoiding types will meet at this point. (not visible because the bins are not fine enough)
This is supported from Ref. Claussen et al. (2015) which shows that the distribution for standard random walks with Gaussian jumps, i.e., without lattice or fixed step length, do not bend down and have an exponential right tail. We conclude that the deviation from this are thus caused by this difference.
First we will look at the rescaled means , where is an observable and its effective dimension, as introduced above in Eq. (5). The scaling is a combination of the scaling of the end-to-end distance and the typical scaling that a -dimensional observable scales as with a characteristic length .
Nevertheless, due to finite-size corrections, the ratios will still depend on the walk length. Thus, the measured estimates at specific walk lengths need to be extrapolated to get an estimate of the asymptotic value . For the extrapolation we use Schawe et al. (2017)
This choice is motivated by a large expansion for the area () of the convex hulls of standard random walks () with Gaussian jumps Grebenkov et al. (2017)
where the constant . A natural guess for a generalization to oberservables of a different effective dimension Schawe et al. (2017) and different walk types would be a similar behavior with different coefficients like Eq. (7).
Indeed, using this form to estimate the asymptotic means of the observable yields good fits, as visible in Fig. 6. In fact, for the fit quality we obtain values between and (the fit ranges for SKSAW begin at , for LRW, SAW and LERW at , hinting at more severe corrections to scaling for the former). We assume that the scaling is thus valid for arbitrary random walk types. The resulting fit parameters are shown in Table 1.
For standard random walks with Gaussian jumps the asymptotic means are known Eldan (2014). These results can be used to predict the corresponding values for LRW. First consider the following heuristic argument for a square lattice. On average a random walk takes the same amount of steps in and direction, such that on average two steps displace the walker by , i.e., the diagonal of a square. In contrast a Gaussian walker with variance will be displaced on average by every step. To make both types comparable, we can increase the lattice constant to , which leads to an average displacement of per step for the LRW. Using the same argumentation for higher dimensions, we can use the trivial scaling with the lattice constant and the length of the diagonal of a unit hypercube , to derive a general conversion:
These known results are listed next to our measurements in Table 1 and are within errorbars compatible with our measurements.
Since we have data for the whole distributions, a natural question is, whether this scaling does apply over the whole support of the distribution. There is evidence that this is true for the convex hulls of standard random walks Claussen et al. (2015) in arbitrary dimensions Schawe et al. (2017). That means the distributions of an observable for different walk lengths should collapse onto one universal function
Fig. 7 shows the distributions of the area of all considered random walk types scaled according to Eq. (10). The curves collapse well in the peak region and in the intermediate right tail. In the far right tail clear deviations from a universal curve are obvious, which are the mentioned finite-size effects caused by the lattice.
The distributions look qualitatively similar, though with weaker finite size effects, i.e., a better collapse, for the perimeter (not shown). In , where we have studied the volume, the results also look similar but exhibit stronger finite-size effects (not shown).
Using the full distributions at different values of the walk length , we can test if it obeys the large deviation principle, i.e., if exists, such that the distribution scales as
for large values of Touchette (2009). To simplify comparison, the support of the rate function is usually normalized to . Here we achieve this by using the maximum Eq. (5). Solving Eq. (11) for results in
We plot this for a selection of our results in Fig. 8. From these plots, seems to approximately follow a power law in the intermediate right tail, while the finite-size effects caused by the lattice play a major role in the far right tail, which “bends up” consequently.
Assuming that the rate function behaves approximately as a power law, which seems consistent with our data shown in Fig. 8, i.e.,
The term on the right-hand side can be ignored next to the exponential. Since the right-hand side is a function of , the left-hand side must also be only dependent on . This can be achieved by assuming , as one can easily see by using Eq. (13):
|Starting from the left-hand side|
From this we can conclude
which simplifies to the case of the standard random walk above the critical dimension of the given walk type Schawe et al. (2017)
To compare this crude estimate with the results of our simulations, we do a point-wise extrapolation of the empirical rate functions for fixed walk lengths as done before in Claussen et al. (2015); Dewenter et al. (2016); Schawe et al. (2017). For the point-wise extrapolation, we use measurements for multiple values of the walk length at fixed values of . Since our data are discrete due to binning, the values of are obtained by cubic spline interpolation. With these data points, which can be thought of as vertical slices through the plots of Fig. 8, we extrapolate the case with a fit to a power law with offset
The extrapolated values are marked with black dots in Fig. 8. Since finite-size effects have major impact on the tails due to the lattice structure, we expect that our estimate is only valid for the intermediate right tail of our simulational data. To estimate sensible uncertainties, we fit different ranges of our data and give the center of the range of the obtained as our estimate with an error including the extremes of the obtained . The black lines in Fig. 8 are our expected values, which are in all examples compatible with some range of our extrapolated data.
In Fig. 8(a) the LRW is shown, which is equivalent to Brownian motion in the large limit for which Ref. Claussen et al. (2015); Schawe et al. (2017) showed the rate function to behave like a power law with exponent for the area in . Using the above mentioned procedure we obtain which is in perfect agreement with the expectation .
Fig. 8(b) shows the same for the SKSAW. The obtained asymptotic rate function’s exponent is compatible with our expectation, though the stronger finite-size effects, lead to larger uncertainties of our estimate.
Fig. 8(c) shows the same but for the volume of the SAW in dimensions. The finite-size effects are apparently stronger for the volume in , as the slope of the right-tail rate function gets less steep with increasing system size.
Fig. 8(d) for the perimeter of a dimensional LERW. In contrast to the area and volume the far right tail of the perimeter seems to bend down instead of up, albeit slightly. Though in the intermediate right tail, the rate function seems to behave as expected.
|Eq. (15)||Eq. (15)|
In general, our data supports the convergence to a limiting rate function, which, mathematically speaking, means that the large-deviation principle holds. This means that the distributions are somehow well behaved and might be accessible to analytical calculations. Though the estimate for what the rate function actually is, can possibly be improved. However, since our estimate for is always compatible with our measurements it appears plausible that also for interacting walks the distribution of the convex hulls is governed by the scaling behavior of the end-to-end distance, as given by the exponents .
We numerically studied the area and perimeter of the convex hulls of different types of self-avoiding random walks in the plane and to a lesser degree the volume of their convex hulls in dimensional space. By applying sophisticated large-deviation algorithms, we calculated the full distributions, down to extremely small probabilities like . We also obtained corresponding rate functions of these observables. Our data support a convergence of the rate functions, which means the large-deviation principle seems to hold. We observed a generalized scaling behavior, which was before established for standard random walks. Thus, although the self-avoiding types of walk exhibit a more complicated behavior as compared to standard random lattice walks, and although the limiting scaled distributions of their convex hull’s volume and surface look quite different for the various walk cases, in the end the convex hull behavior seems to be still governed by the single end-to-end distance scaling exponent .
We also observed, rather expectedly, that the two observables area and perimter are highly correlated for small values. For large but not too large values of the perimeter, many different values of the area are possible, but statistically dominated by rather small values of the area. Extremly large values of the perimeter are only feasible with shrinking area.
Finally, we gave estimates for the large asymptotic mean values of the mentioned observables. These might be of interest for attempts to calculate these values analytically.
For future studies it could be interesting to look closer into the correlations between different observables that we briefly noticed. For a more throughout study, it would be useful to obtain full two-dimensional histograms.
Acknowledgements.This work was supported by the German Science Foundation (DFG) through the grant HA 3169/8-1. HS and AKH thank the LPTMS for hospitality and financial support during one and two-month visits, respectively, where considerable part of the projects were performed. The simulations were performed at the HPC clusters HERO and CARL, both located at the University of Oldenburg (Germany) and funded by the DFG through its Major Research Instrumentation Programme (INST 184/108-1 FUGG and INST 184/157-1 FUGG) and the Ministry of Science and Culture (MWK) of the Lower Saxony State. We also thank the GWDG Göttingen for providing computational resources.
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