COINCIDENCES IN POISSON PATTERNS 1027 



1^.6 An Exact Calculation 



The upper and lower bounds in Section 4.5 are not very close, largely 

 because of the small size of the hexagonal cells. An improved upper 

 bound may be obtained using square cells of side 26. We can calculate 

 p for small rectangular cells but only if we redefine our notion of coin- 

 cidence in terms of square neighborhoods instead of circular neighbor- 

 hoods. That is, points (xi , yi) and (.r2 , 7/2) are now considered coincident 

 if simultaneously 



I xi — X2\ ^ 8, and | ^1 — ^2 | ^5. 



The result we get is the only exact calculation of a non-trivial multi- 

 dimensional coincidence probability known to us. 



Consider the rectangle 0^x^L,0^y^M with L and M both ^ 

 26. If L is less than 5, two points are coincident if and only if their //-co- 

 ordinates differ by less than 6. The problem then reduces to a one-di- 

 mensional coincidence computation such as we gave in Part IL There- 

 fore, suppose both L and M are greater than 6. 



There is probability 



(XLMf -xz,./ 



that the rectangle contains k points. We therefore subdivide the problem 

 into cases of the form "given k, find the probability that the A- points 

 have no coincidences". Only five of these cases have a non-zero answer. 

 To show this, divide the rectangle into four rectangles of sides L/2, M/2; 

 if /o ^ 5 one of these rectangles must contain more than one point, and 

 so a coincidence. The remaining cases k = 0, 1, 2, 3, 4 may be further 

 subdivided according to which pairs of .r-coordinates are less than 5 

 apart. Let us number the k points (.ri , yi), • • •, (.ta; , yk) in such a way 

 that the .r-coordinates are in order Xi ^ x^ S^ • • • ^ .t^ . If, for some i, 

 Xi+2 ^ Xi -f 6, then the subcase in question contributes zero to the 

 probability of no coincidences because all of | .r, — .r,+i [, | Xi+i — .r,:+2 |, 

 I Xi+2 — Xi I are ^ 6 and at least one of | /y, — yi+i |, \ yi+i — yi+o |, 

 I ?/,+2 — yi I is ^6. The only subcases which remain to give a non-zero 

 contribution are the nine listed in Table I. The number in the "subcase" 

 column is k. The next column contains the a--ineciualities which define 

 the subcase. The probability that the k ordered .r-coordinates satisfy the 

 stated inec^ualities is listed as prob^ . If the r-inequalities are satisfied 

 there will be no coincidences if and only if | ^6 — //„ | > 6 for every in- 

 equaUty | Xb — Xa \ ^ 6 given in the a;-inequality column. These y-in- 



