Presumptive Eiruv and the Percolation Transition – Part 2 by Dr. Robert Savit

(2012/5772)

Kol Torah is privileged to reprint a groundbreaking article written by Dr. Robert Savit, an Orthodox physics professor at the University of Michigan with whom Rabbi Jachter worked closely to create the Ann Arbor community Eiruv.  We invite our readers to share their thoughts about this groundbreaking essay which has potential to significantly impact the Eiruvin of our community.

Kol Torah expresses its gratitude to Koninklijke Brille NV publications for permitting us to reprint this important essay, which originally appeared in the journal “Images” volume 5.1 (2011) ISSN 1871-7993, pp.37-43(7).

Last week we discussed the physical concept of the percolation transition and how the probability of a complete path existing increases tremendously based on the probability that short segments in that path exists.  This week we will further the discussion and apply it to community Eiruvim.

III. Presumptive Eiruv

In a highly populated urban area there are many elements that can be combined to form all or part of a Kosher Eiruv. These elements (fences, buildings, properly configured power lines, etc.) are the analogue of the red colored bonds discussed in the last section. Clearly, the greater the population density, the greater the density of Kosher Eiruv elements should be. The density of Kosher elements will also depend on the nature of the urban area—for example, whether some or all of the power lines are underground, as well as the Halachic authority being used to oversee the Eiruv construction.[1] These issues will be briefly addressed in the next section. But for now, it is enough to understand the general analogy. As in the classic percolation problem, the coalescence of (Kosher) elements into an unbroken chain is also clearly necessary for the formation of a Kosher Eiruv. However, the geometric setting for the Eiruv problem is somewhat different than the classic percolation problem discussed in the previous section. Fortunately, an extension of the percolation problem in a much more relevant geometry has also been studied, as we shall now discuss.

 Figure 3a

Figure 3a

 Figure 3b

Figure 3b

Let us introduce a highly simplified model of a city containing a Jewish area, around which we wish to build an Eiruv. We consider again a square grid of linear dimension, L, as in Fig. 1 (editor’s note: See last week’s issue for Figures 1a and 1b). This will be our city. We now suppose that there is a (square) Jewish area at the center of the city whose linear dimension is R, as shown in Figs. 3a and 3b (editor’s note: Due to ink constraints, the red bonds are printed here as black. For color figures, please visit our website). In order to have an Eiruv around this Jewish neighborhood, we want to construct a closed circuit of Kosher Eiruv elements anywhere in the square grid, so long as no portion of the closed circuit penetrates the Jewish neighborhood.

 Figure 4

Figure 4

As in the previous section, we consider randomly coloring red, with probability p, some of the bonds that are in the square grid (the city), but that are not in the Jewish neighborhood. Clearly, if too few of the bonds are colored red, we will not be able to find a closed circuit in the city that surrounds the Jewish neighborhood. On the other hand if p is one, then there will be a very large number of such closed circuits. We now ask whether there is a value of p (call it p*), less than one, at which there is a transition from a low probability of finding a closed circuit (Eiruv) to a probability close to one, for finding at least one such circuit. As an example, we show in Fig. 3a our model city with p=0.4, and in Fig. 3b, our model city with p=0.6. Note that there is no Kosher Eiruv in Fig. 3a, but there is in Fig. 3b. It turns out that for this geometry a curve similar to that shown in Fig. 2 is obtained. I.e., there is a p* such that for p less than p* there is a very small probability of finding a closed circuit, while for p greater than p* the probability of finding at least one closed circuit in the city (i.e. at least one Kosher Eiruv) is close to one, as shown in Fig. 2.[2]          

IV. Practical Considerations and Overarching Halachic Questions

The model of Fig. 3 works well as a theoretical physicist’s version of an Eiruv in a city, and therefore, works well as a proof of concept. But there are a number of important elements missing from this model, some of which may strongly affect the application of the idea of assuming the existence of an Eiruv in a specific real situation. In addition, the entire idea of presumptive Eiruv raises a set of interesting Halachic questions. First, we will discuss some ways in which the details of a specific Eiruv situation could materially alter the typical results shown in Fig. 2. This discussion is not exhaustive, but raises some of the most obvious issues. We will then briefly mention some Halachic questions that the idea of presumptive Eiruv raises.

A. Practical considerations

Even if we accept the basic notion of presumptive Eiruv, some features commonly found in cities may materially affect the application of the model of Fig. 3. Before discussing these issues, it is important to point out that none of them rules out the possibility of presumptive Eiruv. Rather, these are details that may affect the estimate of p, and the estimate of p* for that specific situation. A more elaborate model, similar to Fig. 3 but incorporating the particular relevant details of a specific situation can then be used as the basis for a set of computer simulations in which one can explicitly calculate the probability of finding a Kosher Eiruv without actually constructing one. To put it another way, the structure of Fig. 2 will generally apply, but the details of a specific situation will determine whether the effective value of p is less than or greater than p*, and if it is greater, whether the system (the city) is large enough to warrant a reasonable presumption that an Eiruv exists, absent its specific identification.

1.     General determinants of p.  A number of considerations will affect the estimate of p. We expect that the larger the population density (up to a point) the greater p will be, since there will be a greater density of elements (wires, fences, etc.) that can be incorporated into an Eiruv. The particular Halachic approach will also affect the value of p. Some Halachic authorities will accept certain elements that are rejected by others. For example, the way in which a wire is attached to a pole may be acceptable by some authorities, but not by others. The general nature of construction in the city will also affect the estimate of p. For example, if all the power lines are underground, p may be small, even if the population density is high.

2.     Changes in geometry. It may happen that the city or the Jewish neighborhood is not a square, or that the Jewish neighborhood is not in the center of the city. It may also happen that the city is bounded by a large natural boundary which can form part of an Eiruv. So, a city at the edge of a lake may be able to use that lake shore as part of its Eiruv. In that case, we will not require a complete closed circuit, but only a path that begins and ends on the lake shore and surrounds the Jewish neighborhood. These general geometric considerations will affect the estimate of p*.

3.     Changes to the grid. It may be that a square grid is not a good model for the ways in which the potential Eiruv elements align themselves. This will be particularly true if the city is not laid out on a grid. An additional consideration in this category is that the potential Eiruv elements may not, typically, be of the same length. In a particular situation, a better model might be to allow the red colored bonds to be of different lengths. So, for example, shorter bonds could be associated with buildings, while longer bonds could be associated with long runs of appropriate power lines.

4.     Inhomogeneities. The model of Fig. 3 is homogeneous in that the grid is the same all over the city. But this may not be a good model for many cities. If, for example, freeways or large boulevards cross the city, there may be necessary interruptions in potential Eiruv paths. These will have to be incorporated in a more detailed model relevant to that particular city.

5.     Distribution of Eiruv elements. The simple models discussed here have Kosher Eiruv elements randomly distributed on the associated grid. In a real city, though, the positions of Kosher Eiruv elements may be spatially correlated. That is, it may be that Kosher elements are clustered together, rather than being randomly distributed (for example, because of the way houses or power lines were initially constructed.) This non-random clustering may or may not strongly affect the estimate of p.

6.     Halachic limitations on the size of the Eiruv. An Eiruv, as it is commonly understood, works only in a Karmelit (area in whch it is forbidden to carry only on a rabbinic level), not in a Reshut HaRabim (major thoroughfare), since the legal fictions it embodies can only be used to circumvent a Rabbinic prohibition. If the city is sufficiently dense and busy, it is possible that the necessity of limiting the Eiruv to a Karmelit might impose a restriction on the size of the Eiruv, which would require the linear extent of the Eiruv to be substantially smaller than the size of the city. This would limit the steepness of the curve at p* in Fig. 2, and thereby limit the probability for a presumptive Eiruv, even if p is greater than p*.

B. Overarching Halachic Questions

As described in the last sub-section, Halachic considerations can affect some practical aspects of presumptive Eiruv, such as the estimate of p. But the idea of presumptive Eiruv also raises a set of more fundamental Halachic questions. Intrinsic to the idea of presumptive Eiruv is the notion of probability. Except in the case of an infinitely large city, we are not able to guarantee, with 100% assurance, the existence of a Kosher Eiruv. We may be able to state that a Kosher Eiruv exists with high probability, and we can even put a probabilistically precise quantitative meaning on that statement, but there will also be some probability, however small, that there is no Eiruv.

Halachah is a body of law used to dealing with probabilities. The notion of doubt, Safeik, appears in many places in Halachic reasoning, and is often taken into account in normative Halachic rulings. In many of those cases, one is faced with a situation where information is incomplete and rulings, nevertheless, must be made taking into account only a probabilistic state. But the concept and application of doubt and probability in Halachah is difficult and subtle. The innovation of presumptive Eiruv presents an interesting challenge to Halachah in that the probability here appears in a rather new way. Consequently, it is unclear to what extent, and under what conditions presumptive Eiruv is Halachically acceptable. The determination of its acceptability will require careful analysis. Here I can only suggest a few of the considerations that will arise in that analysis:

1.     Is the information available in principle? In some cases the Halachah seems to be more lenient in situations in which information that would remove a doubt is unavailable in principle. For example, consider the case in which it is known that a piece of meat came from one of a number of butcher shops, a majority of which are Kosher. Under some circumstances, one may rely on the fact that it is a priori more likely that the meat is Kosher to rule that it is permissible to use it. In this case it is not possible, in principle, to determine from which of the butcher shops the meat came. One might argue that the case of presumptive Eiruv is different since it would be possible, in principle, to determine the route of a Kosher Eiruv. However, the application of the notion of “information available in principle” in the body of Halachah is subtle and a more careful analysis, taking into account other related cases of Halachic rulings, is necessary.

2.     The use of Eiruv, including the legal fictions associated with wires and poles (or Lechis) as putative doorways only works in a Karmelit, since the prohibition against carrying in a Karmelit on Shabbat is only a Rabbinic prohibition, not a Biblical prohibition. There is a general Halachic principle to be lenient in cases of Rabbinic prohibitions. Does the adoption of presumptive Eiruv fall under that general principle?

3.     Although it is possible, in principle, to construct or determine the route of an Eiruv in a city to which one might want to apply the notion of presumptive Eiruv, such determination or construction often involves a great expenditure of financial and human capital. Costs for a community Eiruv are often in the tens of thousands of dollars, or more, and can require many hundreds of man-hours to determine a route and maintain the Eiruv. There are cases in the Halachah in which rulings take into account practical costs. For example, consider the case in which some valuable china dinnerware has been rendered non-Kosher. According to most Halachic opinions, such material cannot generally be made Kosher again. However, if the loss of that dinnerware will entail a significant financial cost, one may, in some circumstances, rely on more lenient minority (but still authoritative) opinions which allow the use of the dinnerware. There may be application of this principle to the question of adoption of presumptive Eiruv, particularly in smaller or less wealthy Jewish communities.

These are just some of the notions that a Halachic analysis of the status of presumptive Eiruv should consider. Whether or not a careful analysis leads to the adoption of the idea of presumptive Eiruv as normative, the analysis itself will certainly be interesting, and, like many innovations, may help to sharpen existing Halachic concepts and approaches.

V. Conclusion

The physical and mathematical basis for presumptive Eiruv is clear. It is in principle possible to accurately estimate the probability of a pre-existing Eiruv, and because the percolation transition is so ubiquitous, it is possible to establish well-defined parameter regions in which there is a high, calculable probability that a Kosher Eiruv exists. If, in a particular situation, there is a priori a high probability that an Eiruv exists, the effort and cost to calculationally establish that fact may be substantially less than the human and financial costs of traditional Eiruv construction and maintenance. Another advantage of presumptive Eiruv inheres in its invisibility. One of the goals of good, traditional Eiruv construction is to make the Eiruv as unobtrusive as possible, by minimizing alterations to the urban landscape and using elements that, as much as possible, blend into existing structures. (The sociological, cultural, and aesthetic reasons for this attitude are interesting, but a discussion thereof goes beyond this paper.) If one is able to establish and rely on a presumptive Eiruv, one will have achieved ultimate unobtrusiveness in that the Eiruv qua Eiruv will be completely invisible since no alterations or emendations to existing structures will have been made. (In fact, the Eiruv’s precise route may be unknown, although it will still enclose the target community.) Of course, there will surely be many situations in which the probability of presumptive Eiruv is not high (cases of p being less than p*), and in those cases one must have recourse to the traditional method of Eiruv construction and maintenance.

Despite its great appeal, the process of having the idea of presumptive Eiruv judged Halachically acceptable by mainstream authorities is non-trivial, nor should it be. The idea of presumptive Eiruv raises a number of Halachic and meta-Halachic questions that will need careful and serious thought. One hopes that this innovation, which could substantially reduce the financial and human capital costs for instituting an Eiruv, will receive an open and rigorous appraisal by rabbinic authorities.

Acknowledgements

I am very grateful to Prof. Robert Ziff and Rabbi Rod Glogower for informative and illuminating discussions, to Rabbi Howard Jachter from whom I learned a great deal about Eiruv, and to Maria Riolo for creating the figures for this paper. I also thank an independent reviewer for several insightful comments.

[1] Different Halachic authorities will allow or disallow different elements as being kosher.

[2] The model in Fig. 3 differs only in inconsequential ways from a system studied by Cardy in J. Cardy, J. Stat. Phys. 125, 1-21 (2006).  See also, R. Ziff, Phys. Rev. E 83, 020107(R) (2011).  The geometry of Fig. 3 is that of an annulus, which turns out to be topologically related to a cylinder open at both ends.  It is in this context that percolation has been studied in this geometry.

 

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Presumptive Eiruv and the Percolation Transition – Part 1 by Dr. Robert Savit