The Expected Total Curvature of Random Polygons
Jason Cantarella, Alexander Y Grosberg, Robert B. Kusner, Clayton Shonkwiler
TThe Expected Total Curvature of Random Polygons
Jason Cantarella, ∗ Alexander Y. Grosberg, † Robert Kusner, ‡ and Clayton Shonkwiler ∗ (Dated: July 18, 2018)We consider the expected value for the total curvature of a random closed polygon. Numericalexperiments have suggested that as the number of edges becomes large, the difference between theexpected total curvature of a random closed polygon and a random open polygon with the samenumber of turning angles approaches a positive constant. We show that this is true for a natural classof probability measures on polygons, and give a formula for the constant in terms of the moments ofthe edgelength distribution.We then consider the symmetric measure on closed polygons of fixed total length constructed byCantarella, Deguchi, and Shonkwiler. For this measure, we are able to prove that the expected valueof total curvature for a closed n -gon is exactly π n + π n n − . As a consequence, we show that atleast / of fixed-length hexagons and / of fixed-length heptagons in R are unknotted.
1. INTRODUCTION
The study of random polygons is a fascinating topic in geometric probability and statisticalphysics. Random polygons provide an effective model for long-chain polymers in solution under“ θ -conditions”. There is an essential distinction between open polygonal “arms”, which are easy toanalyze because each edge is sampled independently, and closed polygons, where the closure con-straint imposes subtle global correlations between edges. The fixed-length open polygons with n edges in R form a manifold which has a codimension 3 submanifold of closed polygons. To studyrandom polygons, we must first fix a probability measure on open polygons and a correspondingcodimension 3 Hausdorff measure on closed polygons. Given these measures, we can then studythe statistical properties of the geometry and topology of polygons in each space.In recent work, two of us (Cantarella and Shonkwiler) [2] presented a new measure on thespace of closed n -gons of fixed length constructed using a map from the Stiefel manifold V ( C n ) of orthonormal 2-frames in complex n -space to the space Pol ( n ) of closed n -gons of length 2. Wecalled this measure the symmetric measure . We computed exact expectations of radius of gyrationand squared chord lengths with respect to the symmetric measure. These are global geometricinvariants of polygons in the sense that they involve edges which are far apart along the polygon.In this paper, we are interested in the total curvature κ , which is the sum of the turning angles ateach vertex of the polygon. This is a local invariant of polygons, in the sense that it is determined ∗ University of Georgia, Mathematics Department, Athens GA † New York University, Physics Department, New York, NY ‡ University of Massachussetts, Mathematics Department, Amherst, MA a r X i v : . [ m a t h . DG ] O c t by pairs of adjacent edges. Though we will only consider total curvature in the present work,our methods should also apply to other local geometric invariants such as total torsion, which isdetermined by triples of adjacent edges.If we sample the edges in an open polygonal arm independently according to a sphericallysymmetric distribution, it is easy to see that the expected turning angle at a vertex is π . Thus theexpected total curvature of an n -edge arm (which has n − vertices) is π ( n − . In a closedpolygon, the edges are not independently sampled due to the closure constraint. Since the closureconstraint involves all the edges, it is reasonable to expect that its effect at any given vertex shouldbecome negligible as n → ∞ . Hence we expect that in a closed polygon, the expected turningangle at a vertex should also approach π . This is true, as we will see below.However, it is not true that lim n →∞ E ( κ, Pol ( n )) − π n = 0 . In 2007, Plunkett et al. [14]numerically sampled random closed equilateral polygons of n edges and found that their totalcurvatures are equal to π n + α ( n ) , where α ( n ) tended towards a constant near . as n → ∞ . In2008, one of us (Grosberg) [4] presented an argument to explain why α ( n ) → π = 1 . . . . for equilateral polygons. Numerical experiments [2] suggested that for the symmetric measure α ( n ) → π . This raised an interesting question: why is the asymptotic value of α ( n ) different forthe two measures?Theorem 1 answers this question. With respect to a probability measure µ satisfying mildhypotheses, the expected total curvature of a closed n -gon is π n + α ( n ) , where α ( n ) → π m m .Here, m and m are first and second moments of edgelength. For random polygons sampledaccording to the symmetric measure of [2], this theorem shows that α ( n ) → π , in agreement withour previous numerical experiment.The main result of this paper is that we are able to go much further for the symmetric measureand obtain an exact formula for the expectation of total curvature on Pol ( n ) . To do so, we showin Theorem 2 that the expectation of any scale-invariant function on Pol ( n ) in the symmetricmeasure is equal to the expectation of that function in a new measure called the Hopf-Gaussianmeasure .The Hopf-Gaussian measure is constructed by applying the Hopf map to a multivariate Gaussiandistribution on quaternionic n -space. Since the coordinatewise Hopf map is a quadratic form on H n (cf. Section 3.1), the coordinates of edges of polygons sampled according to this measure aredifferences of chi-squared variables. Hence, they have Bessel distributions. Using this fact, wedetermine the pdf of the sum of k edges in a random arm in Proposition 17. This enables us to findin Proposition 22 an explicit pdf for a pair of edges sampled from a random closed polygon. Wethen compute the expected value for turning angle by integration in Proposition 24. This gives usour main result (Theorem 15): the expectation of total curvature κ for a random closed n -gon inthe symmetric measure σ is E ( κ ; Pol ( n ) , σ ) = π n + π n n − . This calculation gives us some new insight into these polygon spaces. For example, considerthe old question: what fraction of the space of closed n -gons consists of knotted polygons? It hasbeen proved that the fraction of unknots decreases exponentially quickly to zero in various modelsof random polygons [3, 9, 15]. There are also decades of computational experimentation on thisquestion (cf. [12] for references) which show that for small n , unknots are very common. Fewtheorems are known for specific values of n . Our total curvature theorem allows us to prove that,as measured by the symmetric measure, at least / of the space of fixed-length hexagons in R and / of the space of fixed-length heptagons in R consists of unknotted polygons.These methods open up a number of new avenues for exploration and experimentation. Theability to write an explicit pdf for pairs, triplets, or other collections of edges raises the hopeof computing expectations for other interesting scale-invariant functions, such as total torsion oraverage crossing number. Numerical integration with respect to these pdfs is also an effectivemethod for approximating expected values. This can give significantly better results than averagingover large ensembles of polygons (cf. Section 5). This will aid future research on these polygonspaces by allowing conjectured expectations to be tested to high accuracy.
2. ASYMPTOTIC EXPECTED TOTAL CURVATURE OF POLYGONS
The purpose of this section is to compute the asymptotic expected total curvature of closedrandom polygons under some reasonable hypotheses on the probability measure chosen for polygonspace. We will denote the space of n -edge open polygons (up to translation) in R d by A d ( n ) andthe subspace of n -edge closed polygons in R d by P d ( n ) . Here we do not fix the lengths of thepolygons, so A d ( n ) consists of vectors of edges ( (cid:126)e , . . . , (cid:126)e n ) ∈ R d × · · · × R d = R dn and P d ( n ) is the linear codimension d subspace of Arm d ( n ) determined by the closure constraint (cid:80) (cid:126)e i = (cid:126) .We will say that a probability measure µ on A d ( n ) is generated by a spherically symmetric pdf g on R d when µ is the product measure g × · · · × g . The corresponding probability measure on P d ( n ) is the subspace measure with respect to µ . Equivalently, we say that µ is generated by g ifarms are generated by sampling edges independently from g and closed polygons are generated bythe same algorithm conditioned on closure.When the pdf g is spherically symmetric, we can write g ( (cid:126)r ) dVol (cid:126)r = 1Vol S d − | (cid:126)r | d − f ( | (cid:126)r | ) dVol (cid:126)r (1)for some non-negative function f ( r ) so that (cid:82) ∞ f ( r ) d r = 1 (and hence (cid:82) R d g ( (cid:126)r ) dVol (cid:126)r = 1 ).In this case, the radial moments of g are the ordinary moments of f , and both are equal to themoments of edgelength with respect to µ . We denote these by m p := E ( | (cid:126)e i | p ; µ ) = E ( | (cid:126)r | p ; g ) = E ( r p ; f ) . (2)A number of standard probability measures on A d ( n ) are generated in this way. For instance,if f ( r ) = δ ( r − , the resulting measure is the standard measure on n -edge equilateral arms andthe corresponding measure on P d ( n ) is the standard measure on closed equilateral polygons. If f ( r ) = N (0 , , the resulting measure on A d ( n ) is the standard measure on Gaussian random armsand the corresponding measure on P d ( n ) is the standard measure on Gaussian random polygons.If µ is generated by g , it is clear that the expected angle between two edges of a polygon in A d ( n ) sampled according to µ is π , since the edges (cid:126)e i are independently sampled from a sphericallysymmetric pdf on R d . Thus, the expected value of total curvature on A d ( n ) is given by π ( n − .Of course, an n -edge closed polygon in P d ( n ) has an extra turning angle, so we might guess thatthe expectation of total curvature is π n instead. In fact, there is a curvature “surplus” in a closedpolygon. We will now modify the argument in [4] to prove Theorem 1.
For d ≥ , if µ is a measure on A d ( n ) generated by a spherically symmetric pdf g which is bounded on R d and has finite radial moments m , m , and m as in (2) and we take thecorresponding subspace measure on P d ( n ) , then the expected value of total curvature on P d ( n ) approaches π n + dd − d / , d / )B( ( d − / , ( d +1) / ) m m as n → ∞ , where B is the Euler beta function. In particular, when d = 2 and we have E ( κ ; P ( n ) , µ ) (cid:39) π n + 4 π m m and E ( κ ; P ( n ) , µ ) (cid:39) π n + 3 π m m for large n . Proof of Theorem 1.
We will assume for the duration of the proof that A d ( n ) (for any n ) has afixed probability measure µ generated by a fixed spherically symmetric pdf g on R d given by g ( (cid:126)r ) = f ( | (cid:126)r | ) / Vol S d − | (cid:126)r | d − as in (1), and that P d ( n ) has the subspace measure induced by µ .Let the Green’s function G k ( (cid:126)r ) be the probability density of the end-to-end vector (cid:126)r in A d ( k ) with respect to dVol (cid:126)r . We can write this explicitly as G k ( (cid:126)r ) = (cid:90) g ( (cid:126)e ) · · · g ( (cid:126)e k ) δ ( (cid:126)e + · · · + (cid:126)e k − (cid:126)r ) dVol (cid:126)e · · · dVol (cid:126)e k . If we consider the joint probability distribution of all edges in a closed polygon, we can treat itas a conditional probability on a set of edge vectors (cid:126)e , . . . , (cid:126)e n conditioned on the closure constraint (cid:80) (cid:126)e i = 0 . This conditional probability can then be written as P ( (cid:126)e , . . . , (cid:126)e n ) dVol (cid:126)e · · · dVol (cid:126)e n = g ( (cid:126)e ) · · · g ( (cid:126)e n ) δ ( (cid:126)e + . . . + (cid:126)e n ) C n dVol (cid:126)e · · · dVol (cid:126)e n , where C n = G n ( (cid:126) is the codimension d Hausdorff measure of the closed polygon space P d ( n ) .Given this joint distribution on all the edges, we can integrate out all but two of the edges to getthe joint probability distribution on two consecutive edges, P ( (cid:126)e i , (cid:126)e i +1 ) . Since this is independentof i , we may as well consider the case i = 1 : P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e = g ( (cid:126)e ) g ( (cid:126)e ) G n − ( − (cid:126)e − (cid:126)e ) C n dVol (cid:126)e dVol (cid:126)e . (3)In other words, in order for the edges (cid:126)e and (cid:126)e to come from a closed polygon, the remaining n − edges must connect the head of (cid:126)e to the tail of (cid:126)e .Finding exact expressions for G n − and C n is quite challenging in general, but we can approx-imate both fairly easily by observing that the failure-to-close vector for an element of A d ( k ) isjust the sum of the edges. Given that we are sampling edges of our arms independently, that thethird moment of g is finite, and that g is a bounded density on R d , the vector local limit theorem ofBikjalis [1] implies that the pdf of the (normalized) failure-to-close distribution converges in supnorm to the pdf of a normal distribution.We can recover the parameters of this normal distribution by noting that spherical symmetryimplies that the mean of the failure-to-close distribution is zero, the variance of each coordinate ofan edge vector is m / d , and the coordinates of an edge vector are uncorrelated. Therefore, the pdfof √ k G k ( (cid:126)r ) converges in sup norm to the pdf of the d -dimensional normal distribution N (cid:16) (cid:126) , diag ( m / d , . . . , m / d ) (cid:17) , where diag( a , . . . , a k ) is the diagonal k × k matrix with entries a , . . . , a k .In particular, as n → ∞ , we have that G n − and C n = G n ( (cid:126) are asymptotic in sup norm to G n − ( (cid:126)r ) (cid:39) (cid:18) d π ( n − m (cid:19) d / exp (cid:18) − dr n − m (cid:19) C n (cid:39) (cid:18) d π n m (cid:19) d / , where r = | (cid:126)r | . From (3), then, we see that the pdf P ( (cid:126)e , (cid:126)e ) is sup norm close to the function P ( (cid:126)e , (cid:126)e ) (cid:39) g ( (cid:126)e ) g ( (cid:126)e ) (cid:18) nn − (cid:19) d / exp (cid:18) − d | (cid:126)e + (cid:126)e | n − m (cid:19) . (4)Let θ ( (cid:126)e , (cid:126)e ) be the angle between (cid:126)e and (cid:126)e , which is to say the turning angle between the twoedges. We will now prove that E ( θ ) → π + dn ( d −
1) B( d / , d / )B( ( d − / , ( d +1) / ) m m as n → ∞ .First, for any (cid:15) > we may choose ρ so that the integral of θ ( (cid:126)e , (cid:126)e ) P ( (cid:126)e , (cid:126)e ) over the comple-ment of the ball B ( ρ ) of radius ρ centered at the origin obeys (cid:90) R d − B ( ρ ) θ ( (cid:126)e , (cid:126)e ) P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e < (cid:15). (5)To see this, observe that P ( (cid:126)e , (cid:126)e ) is a pdf on R d , so its improper integral over the entire spaceconverges. This means that the L norm of P on the complement of a ball of radius ρ goes to as ρ → ∞ . But θ is bounded by π so (cid:107) θ ( (cid:126)e , (cid:126)e ) P ( (cid:126)e , (cid:126)e ) (cid:107) ≤ π (cid:107) P ( (cid:126)e , (cid:126)e ) (cid:107) where the norms areover the complement of the ball B ( ρ ) . Choosing ρ large enough that the rhs is less than (cid:15) yields (5).Similarly, g ( (cid:126)e ) g ( (cid:126)e ) is the pdf of a two-edge arm, so its improper integral over R d convergesas well. Since g has finite first and second moments on R d , this product has finite mixed momentsof order up to 2 on R d . In particular, for any quadratic polynomial q ( r , r ) with coefficientsbounded by ± λ we may choose ρ ( λ ) so that we have (cid:90) R d − B ( ρ ) q ( r , r ) g ( (cid:126)e ) g ( (cid:126)e ) dVol (cid:126)e dVol (cid:126)e < (cid:15). (6)We now turn to the interior of the ball. Since the pair ( (cid:126)e , (cid:126)e ) is in the interior of B ( ρ ) in R d , wemay choose n large enough that | (cid:126)e + (cid:126)e | ( n − m is as close to zero as we like. In particular, we may choose n large enough that the exponential in (4) is sup norm close to its linear Taylor approximation on theentire ball. Further, we can approximate (cid:16) nn − (cid:17) d / by dn and n − by n and we have P ( (cid:126)e , (cid:126)e ) sup norm close to the following function over the entire ball: P ( (cid:126)e , (cid:126)e ) (cid:39) g ( (cid:126)e ) g ( (cid:126)e ) (cid:20) dn − d | (cid:126)e | nm − d | (cid:126)e | nm − d (cid:104) (cid:126)e , (cid:126)e (cid:105) nm (cid:21) . (7)The expected value of θ ( (cid:126)e , (cid:126)e ) is just (cid:90) R d θ ( (cid:126)e , (cid:126)e ) P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e = (cid:90) B ( ρ ) θ ( (cid:126)e , (cid:126)e ) P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e + (cid:90) R d − B ( ρ ) θ ( (cid:126)e , (cid:126)e ) P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e . (8)By (5), the second integral on the right is small and we can ignore it. Since the ball is a boundeddomain, the fact that P ( (cid:126)e , (cid:126)e ) is sup norm close to the approximation in (7) tells us that the integralof the bounded function θ against the approximation is close to the first integral on the right.Consider the approximation (7) to P ( (cid:126)e , (cid:126)e ) . For large enough n , the quantity in square bracketsis a quadratic polynomial in r and r with coefficients between − and , so the inequality (6)applies with ρ ≥ ρ ( √ . Hence, since θ is bounded, its integral against the approximation overthe ball is close to its integral against the approximation over all of R d . In other words, we canapproximate the first integral on the rhs of (8) by the integral (cid:90) R d θ ( (cid:126)e , (cid:126)e ) g ( (cid:126)e ) g ( (cid:126)e ) (cid:20) dn − d | (cid:126)e | nm − d | (cid:126)e | nm − d (cid:104) (cid:126)e , (cid:126)e (cid:105) nm (cid:21) dVol (cid:126)e dVol (cid:126)e . We now evaluate the above integral. We will write (cid:126)e and (cid:126)e in spherical coordinates. Sincethe integrand is spherically symmetric, we can integrate out the angular coordinates of, say, (cid:126)e and assume that (cid:126)e lies along the z -axis. This produces a factor of Vol S d − . Since the rotationof (cid:126)e in the ( d − -plane perpendicular to the (cid:126)e -axis does not change θ or the approximationto P ( (cid:126)e , (cid:126)e ) , we can integrate out another Vol S d − . Since θ is now the polar angle for (cid:126)e and Vol S d − Vol S d − = Γ( d / ) √ π Γ( ( d − / ) , the integral reduces to Γ ( d / ) √ π Γ ( ( d − / ) π (cid:90) ∞ (cid:90) ∞ (cid:90) θ f ( r ) r d − f ( r ) r d − (cid:20) dn − dr nm − dr nm − dr r nm cos θ (cid:21) r d − r d − sin d − θ d r d r d θ. Since (cid:82) ∞ f ( r i ) d r i = 1 and (cid:82) ∞ r pi f ( r i ) d r i = m p , integrating out r and r yields Γ ( d / ) √ π Γ ( ( d − / ) (cid:90) π θ (cid:18) − dm nm cos θ (cid:19) sin d − θ d θ = π dn ( d −
1) B ( d / , d / )B ( ( d − / , ( d +1) / ) m m after integrating by parts.Since this is the expected value of the turning angle between two edges of the polygon, multi-plying by n yields the desired expression for expected total curvature.
3. THE SYMMETRIC AND HOPF-GAUSSIAN MEASURES ON POLYGON SPACES
Theorem 1 applies to a broad class of measures on polygon space, but not to certain highlysymmetric measures defined in [2]. These symmetric measures on fixed-length polygons in spaceand in the plane are interesting for a number of reasons: they come from a natural geometric con-struction, expectations and moments of chordlengths and radii of gyration are exactly computableand scale like the corresponding expectations for equilateral polygons, and there is an algorithmfor direct sampling from these measures which is fast (linear in the number of edges) and easy tocode. Our goal for the rest of the paper is to determine the expected total curvature of polygonswith respect to these measures.The symmetric measure is most naturally defined on the space of (open or closed) n -gons offixed total length in either R or R , which we denote by Arm d ( n ) for open polygons and Pol d ( n ) for closed polygons in R d . Of course we can extend the definition to the space of polygons of anyfixed length by scaling. Viewed as a subspace of A d ( n ) , the space Pol d ( n ) differs from P d ( n ) inthat elements satisfy constraints on both closure and total length. Therefore, Theorem 1 does notapply to the symmetric measure.However, since total curvature is a scale-invariant quantity and since P d ( n ) is a cone over Pol d ( n ) , the expected total curvature of polygons in P d ( n ) – which we can determine asymptot-ically using Theorem 1 – will be the same as the expected total curvature of polygons in Pol d ( n ) provided that this expectation is computed with respect to a measure on P d ( n ) which is a prod-uct of some measure on the cone parameter and the symmetric measure on Pol d ( n ) . Indeed, inSection 3.2 we will define the Hopf-Gaussian measure H on A d ( n ) and P d ( n ) for d = 2 , byapplying the Hopf map to the standard multivariate Gaussian measure on H n . The Hopf-Gaussianmeasure on A d ( n ) turns out to be the product H = χ d − n × σ , where χ d − n is the chi-squareddistribution with d − n degrees of freedom on the interval [0 , + ∞ ) which parametrizes the conedirection and σ is the symmetric measure on Arm d ( n ) . Likewise, the Hopf-Gaussian measure on P d ( n ) is the product H = χ d − n × σ , where now σ is the symmetric measure on Pol d ( n ) .An immediate consequence of this construction is the following theorem, which is the centralmessage of this section: Theorem 2.
Suppose F : A d ( n ) → R is a scale-invariant function. Then the expected value of F over A d ( n ) with respect to the Hopf-Gaussian measure H is the same as the expected value of F over Arm d ( n ) with respect to the symmetric measure σ ; that is E ( F ; A d ( n ) , H) = E ( F ; Arm d ( n ) , σ ) . Likewise, if F : P d ( n ) → R is scale-invariant, then E ( F ; P d ( n ) , H) = E ( F ; Pol d ( n ) , σ ) . As we will see, the Hopf-Gaussian measure satisfies the hypotheses of Theorem 1, so the com-bination of Theorems 1 and 2 will allow us to determine the expected asymptotic total curvature on
Pol d ( n ) with respect to σ from the first and second moments of edgelength on A d ( n ) with respectto H , which we compute in Section 3.3. These asymptotic total curvature expectations are givenby: Corollary 3.
For d ∈ { , } and large n , the expected total curvature on Pol d ( n ) with respect tothe symmetric measure is E ( κ ; Pol ( n ) , σ ) (cid:39) π n + 2 π , E ( κ ; Pol ( n ) , σ ) (cid:39) π n + π . The value of π for the total curvature surplus of polygons in R agrees with our numericalexperiments in [2].Of course, Theorem 2 applies to any scale-invariant functional on polygons, not just total cur-vature. We expect that it will be useful for determining the expected values of other interestingquantities such as total torsion and average crossing number. In this subsection we recall the construction of the symmetric measure on
Arm d ( n ) and Pol d ( n ) from [2]. Recall that these are spaces of arms and polygons of fixed total length 2. In principleeverything could be scaled to any desired fixed length, but the choice of length 2 will be the mostconvenient. Since the translation of the following definitions and results to any other scale isstraightforward, we will not discuss this scaling further. Definition 4.
For d ∈ { , } , let Arm d ( n ) be the moduli space of n -edge polygonal arms (whichmay not be closed) of length 2 up to translation in R d . An element of Arm d ( n ) is a list of edgevectors (cid:126)e , . . . , (cid:126)e n ∈ R d whose lengths sum to .Consider the Hopf map from the division algebra of quaternions H to the space of imaginaryquaternions (which we identify with R ) given by Hopf( q ) = ¯ q i q, where ¯ q is the quaternionic conjugate of q . In coordinates, if q = ( q , q , q , q ) , then Hopf( q ) = ( q + q − q − q , q q − q q , q q + 2 q q ) . (9)We extend the Hopf map coordinatewise to a map Hopf : H n → R n . Then Hopf is a smooth mapfrom the sphere S n − of radius √ in H n onto Arm ( n ) . Specifically, for (cid:126)q ∈ S n − the edge setof the polygon Hopf( (cid:126)q ) is ( (cid:126)e , . . . , (cid:126)e n ) := (Hopf( q ) , . . . , Hopf( q n )) . We call S n − the model space for Arm ( n ) .Similarly, the restriction of Hopf to the ⊕ j and i ⊕ k planes gives a map to the i ⊕ k plane,which we identify with C = R . Specifically, Hopf( a + b j ) = i ( a + b j ) Hopf( a i + b k ) = i ( a + b j ) . In other words, if z = a + b j , then Hopf( z ) = i ¯ z and Hopf( i z ) = i z . Extending this mapcoordinatewise yields a smooth, surjective map Hopf : S n − (cid:116) S n − → Arm ( n ) ; consequently,the disjoint union S n − (cid:116) S n − is the model space for Arm ( n ) . Definition 5.
For d ∈ { , } , let Pol d ( n ) be the moduli space of closed n -gons of length 2 up totranslation in R d .0Since Pol d ( n ) ⊂ Arm d ( n ) , the inverse image Hopf − (Pol d ( n )) is well-defined and will be themodel space for Pol d ( n ) . To describe this model space for d = 3 as a subset of S n − , the sphereof radius √ in H n , it is convenient to write S n − as the join S n − (cid:63) S n − . In coordinates, thejoin map is given by ( (cid:126)u, (cid:126)v, θ ) (cid:55)→ √ θ(cid:126)u + sin θ(cid:126)v j ) where (cid:126)u, (cid:126)v ∈ C n lie on the unit sphere and θ ∈ [0 , π / ] . The Stiefel manifold V ( C n ) of Hermitianorthonormal 2-frames ( (cid:126)u, (cid:126)v ) in C n can be identified with the subspace { ( (cid:126)u, (cid:126)v, π / ) : (cid:104) (cid:126)u, (cid:126)v (cid:105) = 0 } ⊂ S n − and, as Hausmann and Knutson first observed [5], this manifold is precisely the model space for Pol ( n ) . Proposition 6 ([5]) . The coordinatewise Hopf map takes V ( C n ) ⊂ C n × C n = H n onto Pol ( n ) . The key to proving the above proposition is to note that the Hopf map applied to a quaternion q can be written more simply by letting q = a + b j for a, b ∈ C : Hopf( q ) = Hopf( a + b j ) = ( | a | − | b | , (cid:61) ( a ¯ b ) , (cid:60) ( a ¯ b )) = i ( | a | − | b | + 2 a ¯ b j ) . Let V ( R n ) be the real Stiefel manifold of orthonormal 2-frames in R n , which sits naturally in R n ⊕ j R n . A result analogous to the above holds for planar polygons: Proposition 7 ([5]) . The coordinatewise Hopf map takes V ( R n ) (cid:116) i V ( R n ) onto Pol ( n ) . With these maps in place, we can define probability measures on the arm and polygon spaces bypushing forward measures on the model spaces. Since the model spaces are homogeneous spaces,it is natural to push forward Haar measure on the model spaces, which is what we did in [2]. SinceHaar measure is also the measure defined by the standard Riemannian metrics on these spaces, thisgives the following definition of the symmetric measure σ on Pol ( n ) : σ ( U ) = 1Vol V ( C n ) (cid:90) Hopf − ( U ) d Vol V ( C n ) for U ⊂ Pol ( n ) . The symmetric measures on the other arm and polygon spaces are defined analogously.The space
Pol d ( n ) is topologically the union of the spaces of polygons with fixed edgelengths r , . . . , r n such that (cid:80) r i = 2 , so any expectation over Pol d ( n ) with respect to the symmetricmeasure is a weighted average of the expectations over these spaces. In future work we intend todetermine how the average is weighted and with respect to which measure on the fixed edgelengthspaces. For equilateral polygons the answer is simple and pleasant: the restriction of the symmetricmeasure to the subspace of equilateral polygons is just the natural measure on this space, namelythe subspace measure on n -tuples of vectors in the round S which sum to zero.1 We would like to compute the expectation of total curvature over
Pol d ( n ) with respect to thesymmetric measure. Unfortunately, since elements of Pol d ( n ) are chosen from A d ( n ) by con-ditioning on the polygon both being closed and having total length 2, we cannot directly applyTheorem 1. However, since the total curvature κ is scale-invariant and since P d ( n ) is a cone over Pol d ( n ) (with the cone direction parametrized by the length of the polygon), we have that E ( κ ; P d ( n ) , µ ) = E ( κ ; Pol d ( n ) , σ ) for any measure µ on P d ( n ) such that µ = ρ × σ for some measure ρ on [0 , + ∞ ) .At the level of model spaces, the picture is clearer. Hopf maps H n onto A ( n ) and the imageof the sphere of radius r is exactly the copy of Arm d ( n ) consisting of polygonal arms with totallength r . Moreover, the measure on this scaled copy of Arm d ( n ) is exactly the pushforward of thestandard measure on the sphere of radius r , so it is the symmetric measure defined in the previoussection. Therefore, we can define a measure on A ( n ) which is the product of some measureon [0 , + ∞ ) and the symmetric measure on Arm ( n ) simply by pushing forward any sphericallysymmetric measure on H n . Of course, if we may choose any spherically symmetric measure,the obvious choice is the multivariate Gaussian measure: this is both a spherically symmetricmeasure and a product measure on the coordinates and we can expect that the fact that the individualcoordinate distributions are Gaussian will simplify our computations considerably.In the case of planar polygons, the model space for A ( n ) is the explicit copy of C n ∪ C n givenby ( R n ⊕ j R n ) ∪ ( i R n ⊕ k R n ) , so we will push forward the Gaussian measure on C n : Definition 8. If γ n is the standard Gaussian measure on H n = R n , then the Hopf-Gaussianmeasure H on A ( n ) is defined by H( U ) = (cid:90) Hopf − ( U ) dγ n for U ⊂ A ( n ) . Likewise, if γ n is the measure on ( R n ⊕ j R n ) ∪ ( i R n ⊕ k R n ) naturally induced by the standardGaussian measure on C n = R n , then the Hopf-Gaussian measure H on A ( n ) is defined by H( U ) = (cid:90) Hopf − ( U ) dγ n for U ⊂ A ( n ) . The fact that the multivariate Gaussian is a product measure implies that:2
Proposition 9.
The Hopf-Gaussian measure on A ( n ) is generated by the pdf g ( (cid:126)r ) = e − | (cid:126)r | / π | (cid:126)r | and the Hopf-Gaussian measure on A ( n ) is generated by the pdf g ( (cid:126)r ) = e − | (cid:126)r | / π | (cid:126)r | . Proof.
Since γ n is a product measure on H n , its restriction to each H factor is the standard four-dimensional Gaussian. In particular, for (cid:126)q = ( q , . . . , q n ) ∈ H n sampled according to γ n , the q i ∈ H are independent, identically distributed, and spherically symmetric. Therefore, the edges ofthe polygon Hopf( (cid:126)q ) = (Hopf( q ) , . . . , Hopf( q n )) are independent, identically distributed, and,since the Hopf map is SU (2) -equivariant, spherically symmetric. Therefore, the Hopf-Gaussianmeasure H on A ( n ) is generated by a spherically symmetric distribution on R , which we nowdetermine.The four real components of each q i are themselves Gaussian-distributed. Therefore, since | Hopf( q i ) | = | q i | , each edgelength of a Hopf-Gaussian polygon is given by the sum of the squaresof four Gaussian real numbers, so these edgelengths follow a chi-squared distribution with fourdegrees of freedom. Thus, the pdf of the edgelength distribution is f ( r ) = re − r / . But then the pdf g of the spherically symmetric edge distribution is g ( (cid:126)r ) = 14 π | (cid:126)r | f ( | (cid:126)r | ) = e − | (cid:126)r | / π | (cid:126)r | , as desired.The proof in the 2-dimensional case is completely parallel.To identify the model space for P d ( n ) , note that, as with the Pol d ( n ) and Arm d ( n ) spaces,we have that P d ( n ) ⊂ A d ( n ) . Therefore, Hopf − ( P d ( n )) is a well-defined subset of H n whichwill be the model space for P d ( n ) . We can identify this model space more explicitly as follows.Focusing on the case d = 3 for the moment, since H n is the cone over S n − it is convenient towrite H n as the cone of the join S n − (cid:63) S n − . In coordinates, the “cone-join” map is given by ( (cid:126)u, (cid:126)v, θ, s ) (cid:55)→ s (cos θ(cid:126)u + sin θ(cid:126)v j ) where (cid:126)u, (cid:126)v ∈ C n are unit vectors, θ ∈ [0 , π / ] and s ∈ [0 , + ∞ ) . The cone CV ( C n ) over the Stiefelmanifold V ( C n ) can then be identified with the subspace { ( (cid:126)u, (cid:126)v, π / , s ) : (cid:104) (cid:126)u, (cid:126)v (cid:105) = 0 } ⊂ H n and the proof of Proposition 6 generalizes to show that CV ( C n ) is the model space for P ( n ) :3 Proposition 10.
The coordinatewise Hopf map takes CV ( C n ) ⊂ H n onto P ( n ) . If CV ( R n ) is the cone over the real Stiefel manifold V ( R n ) , then the same reasoning yieldsthe analogue of Proposition 7: Proposition 11.
The coordinatewise Hopf map takes CV ( R n ) ∪ i CV ( R n ) ⊂ H n onto P ( n ) . Since P d ( n ) ⊂ A d ( n ) , we can define the Hopf-Gaussian measure on P d ( n ) as the subspacemeasure inherited from the Hopf-Gaussian measure on A d ( n ) from Definition 8.To prove Theorem 2, which says that the expected value of any scale-invariant function onpolygons is the same whether we compute it with respect to the Hopf-Gaussian measure or thesymmetric measure, it suffices to show that the Hopf-Gaussian measure is a product measure: Proposition 12.
Suppose d = 2 or . Then the Hopf-Gaussian measure H on A d ( n ) is the product χ d − n × σ of the chi-squared distribution with d − n degrees of freedom on the interval [0 , + ∞ ) and the symmetric measure on Arm d ( n ) .Likewise, the Hopf-Gaussian measure on P d ( n ) is the product χ d − n × σ of the chi-squareddistribution with d − n degrees of freedom on [0 , + ∞ ) and the symmetric measure on Pol d ( n ) .Proof. Since the Gaussian measure on H n = R n is SO (4 n ) -equivariant, its restriction to thesphere S n − ( r ) of radius r is, after normalization, just the uniform probability measure on thesphere. Since Hopf( S n − ( r )) is the space of arms of total length r , this means that the restric-tion of the Hopf-Gaussian measure on A ( n ) to this space is, after normalization, the symmetricprobability measure σ defined in Section 3.1. Likewise, the restriction of the Hopf-Gaussian mea-sure on A ( n ) to planar arms of total length r is just the symmetric measure.Therefore, the measure on A d ( n ) is the product ρ × σ for some measure ρ on the interval [0 , + ∞ ) , so it suffices to see that ρ is the chi-squared distribution. Since the interval parametrizesthe total length of a polygon, we need to analyze the distribution of total length of polygonal arms.For (cid:126)q = ( q , . . . , q n ) ∈ H n , the arm Hopf( (cid:126)q ) ∈ A ( n ) has total length (cid:88) | Hopf( q i ) | = (cid:88) | ¯ q i i q i | = (cid:88) | q i | . Since each | q i | is the sum of the squares of four standard Gaussians, the total length of the polygon Hopf( (cid:126)q ) follows the standard chi-squared distribution with n degrees of freedom. Therefore, themeasure ρ on [0 , + ∞ ) is the measure induced by the chi-squared distribution with n degrees offreedom. Likewise, for polygonal arms in the plane, the measure on total length is induced by thestandard chi-squared distribution with n degrees of freedom, since in that case each | q i | is thesum of the squares of two standard Gaussians.The fact that the Hopf-Gaussian measure on P d ( n ) is the product of the chi-squared distributionon [0 , + ∞ ) and the symmetric measure on Pol d ( n ) then follows immediately from the definitionof the Hopf-Gaussian measure on P d ( n ) ⊂ A d ( n ) as the subspace measure and the fact that thesymmetric measure on Pol d ( n ) ⊂ Arm d ( n ) is the subspace measure.4Theorem 2 now follows since a scale-invariant function is by definition independent of the firstfactor in the product decomposition of the Hopf-Gaussian measure. By Proposition 9 the Hopf-Gaussian measure on A d ( n ) for d ∈ { , } is generated by thespherically symmetric pdf g ( (cid:126)r ) = e − | (cid:126)r | / d − π | (cid:126)r | . This is certainly a bounded density with finite first, second, and third moments, so we can useTheorem 1 to compute the asymptotic expected total curvature on P d ( n ) with respect to the Hopf-Gaussian measure. To do so, we just need to know the first and second moments of edgelength.In fact, as we saw in the proof of Proposition 9, the edgelength distribution on A d ( n ) is the chi-squared distribution with d − degrees of freedom, so the moments of edgelength are just thewell-known moments of this distribution: Proposition 13.
The p th moment of edgelength on A d ( n ) is given by E ( | e i | p ; A ( n ) , H) = 2 p p ! , E ( | e i | p ; A ( n ) , H) = 2 p ( p + 1)! Note that the expected values of edgelength are E ( | (cid:126)e i | ; A ( n ) , H) = 2 , E ( | (cid:126)e i | ; A ( n ) , H) = 4 (10)and the expected squared edgelengths areand E ( | (cid:126)e i | ; A ( n ) , H) = 8 , E ( | (cid:126)e i | ; A ( n ) , H) = 24 . (11)Using the above values for m and m , Theorem 1 implies that the asymptotic expected totalcurvature on P d ( n ) is given by E ( κ ; P ( n ) , H) (cid:39) π n + 2 π , E ( κ ; P ( n ) , H) (cid:39) π n + π . Since total curvature is scale-invariant, Theorem 2 implies Corollary 3, which says that for large nE ( κ ; Pol ( n ) , σ ) (cid:39) π n + 2 π , E ( κ ; Pol ( n ) , σ ) (cid:39) π n + π . Also, we can now compute the expected value of chordlength and radius of gyration for A d ( n ) using results from [2].5 Corollary 14.
The expected value of the squared length of a chord skipping k edges on A d ( n ) is E (Chord( k ); A ( n ) , H) = 8 k, E (Chord( k ); A ( n ) , H) = 24 k. The expected squared radius of gyration for arms in A d ( n ) is E (Gyradius; A ( n ) , H) = 43 n ( n + 2) n + 1 , E (Gyradius; A ( n ) , H) = 4 n ( n + 2) n + 1 . Proof.
Since our measure on A d ( n ) is invariant under rearrangement of edges, we can easily com-pute expected squared chord length and radius of gyration using Propositions 5.3, 6.3, and 6.5from [2]. Those propositions imply that E (Chord( k ); A d ( n ) , H) = kE ( | (cid:126)e i | ; A d ( n ) , H) (12)and E (Gyradius; A d ( n ) , H) = (cid:18) n ( n + 2)6( n + 1) (cid:19) E ( | (cid:126)e i | ; A d ( n ) , H) . (13)Substituting the second moment of edgelength from (11) into (12) and (13) yields the desiredresults.
4. EXPECTED TOTAL CURVATURE OF POLYGONS WITH THE SYMMETRIC MEASURE
Corollary 3 gave the asymptotic expected total curvatures with respect to the symmetric mea-sures as E ( κ ; Pol ( n ) , σ ) (cid:39) π n + 2 π , E ( κ ; Pol ( n ) , σ ) (cid:39) π n + π . Our aim in this section is to use the special properties of the Hopf-Gaussian measure to carry out theargument of Theorem 1 for space polygons with no approximations. This will yield the followingexact expectation which holds for any n ≥ : Theorem 15.
The expected total curvature with respect to the symmetric measure on
Pol ( n ) andthe Hopf-Gaussian measure on P ( n ) is given by E ( κ ; Pol ( n ) , σ ) = E ( κ ; P ( n ) , H) = π n + π n n − . Since total curvature is scale-invariant, the first equality is a consequence of Theorem 2; provingthe second equality is the main task of this section. Also, since all triangles have total curvature π , Theorem 15 is trivially true for n = 3 . Therefore, throughout the rest of the section we willassume n > .We will shortly be evaluating many definite integrals involving the Bessel function K ν ( z ) ; todo so we will repeatedly avail ourselves of the following:6 Lemma 16.
For real µ > | ν | and α > , (cid:90) ∞ x µ − e − αx K ν ( αx ) d x = √ π µ β µ Γ( µ + ν )Γ( µ − ν )Γ( µ + / ) (14) and (cid:90) ∞ x µ − K ν ( αx ) d x = √ π ν α µ Γ( µ − ν ) µ + ν Γ( µ / + ν / )Γ( µ / + / ) . (15) Proof.
Both equations follow from the identity [7, 6.621(3)] (cid:90) ∞ x µ − e − αx K ν ( βx ) d x = √ π (2 β ) ν ( α + β ) µ + ν Γ( µ + ν )Γ( µ − ν )Γ( µ + / ) F (cid:18) µ + ν, ν + 12 ; µ + 12 ; α − βα + β (cid:19) , which holds for any complex numbers µ, ν, α, β with (cid:60) ( µ ) > |(cid:60) ( ν ) | and (cid:60) ( α + β ) > . Thefunction F is Gauss’s hypergeometric function.To get (14), we simply use the fact that F ( a, b ; c ; 0) = 1 for any a, b, c ; to get (15) we useKummer’s identity: F ( a, b ; a − b + 1; −
1) = Γ( a − b +1)Γ( a / +1)Γ( a +1)Γ( a / − b +1) . k Edges in A ( n ) Since the goal is to carry out the strategy from Section 2 with no approximations, we first needto explicitly determine the Green’s function G k : Proposition 17.
The probability distribution of the vector (cid:126)r joining the ends of a k -edge sub-armof an arm in A ( n ) with the Hopf-Gaussian measure is spherically symmetric in R and given bythe following explicit formula: G k ( (cid:126)r ) dVol (cid:126)r = r k − / K k − / ( r / )2 k +2 π / Γ( k ) dVol (cid:126)r , where r = | (cid:126)r | .Proof. Suppose (cid:126)q = ( q , . . . , q n ) ∈ H k is sampled from the standard Gaussian distribution. Writ-ing q i = a i + b i i + c i j + d k k for each i = 1 , . . . , k , the failure-to-close vector for the k -edge arm Hopf( (cid:126)q ) ∈ A ( n ) is (cid:88) Hopf( q i ) = (cid:88) Hopf( a i + b i i + c i j + d k k ) = (cid:88) ( a i + b i − c i − d i , b i c i − a i d i , a i c i +2 b i d i ) . Again, the fact that this vector follows a spherically symmetric distribution is a consequence of thefact that the Hopf map is SU (2) -equivariant.7Since the a i , b i , c i , d i are chosen from standard (real) Gaussian distributions, the distribution ofthe projection of the failure-to-close vector onto the first coordinate is clearly the difference of twochi-squared distributions, each with k degrees of freedom. This is known to have a pdf given bya Bessel function distribution [8, Chapter 12, Section 4.4] in the form f ( y ) = | y | k − / k √ π Γ( k ) K k − / ( | y | / ) . (16)It is worth noting that this pdf was proved by McLeish [11] to be the pdf of a product of a gamma ( k, variable and an independent standard normal. McLeish also works out the moments andcumulants of the distribution.Next, we will use a result of Lord [10] which relates the pdf of the projection of a sphericallysymmetric distribution to the pdf of the full distribution. Lemma 18 ([10, Eq. (29)]) . Suppose p ( (cid:126)r ) is a spherically symmetric distribution on R and thatthe projection of p ( (cid:126)r ) to any radial line through the origin has the pdf p ( r ) , where r = | (cid:126)r | . Then p ( (cid:126)r ) is given by p ( (cid:126)r ) = − πr p (cid:48) ( r ) dVol (cid:126)r . Using the pdf of the projection given in (16), Lemma 18 implies that the failure-to-close distri-bution is G k ( (cid:126)r ) dVol (cid:126)r = − πr ddr (cid:32) r k − / k √ π Γ( k ) K k − / ( r / ) (cid:33) dVol (cid:126)r This can be re-written using the derivative identity K (cid:48) ν ( u ) = − ( K ν − ( u ) + K ν +1 ( u )) and therecurrence relation K ν ( u ) = K ν − ( u ) + ν − u K ν − ( u ) (cf. [6, 10.29.1] for both) to get thedesired expression G k ( (cid:126)r ) dVol (cid:126)r = r k − / K k − / ( r / )2 k +2 π / Γ( k ) dVol (cid:126)r where r = | (cid:126)r | .Using the identity K − / ( z ) = √ π √ z e − z (cf. [6, 10.39.2]) and specializing Proposition 17 to thecase k = 1 , we see that the distribution of an edge in a Hopf-Gaussian arm is G ( (cid:126)r ) dVol (cid:126)r = e − r / πr dVol (cid:126)r . Note that this is, as it should be, the same distribution for edges that we found in Proposition 9.8 P ( n ) Proposition 19.
The codimension 3 Hausdorff measure of P ( n ) in A ( n ) is the value G n ( (cid:126) ,which is given by C n = Γ( n − / )64 √ π Γ( n ) Proof.
We saw in Proposition 17 that G n ( (cid:126)r ) = 2 − n − r n − / K n − / ( r / ) π / Γ( n ) . (17)We must be careful evaluating this formula at r = 0 , since the Bessel function K n − / has a poleat . To rewrite (17) in a form which allows us to easily evaluate at r = 0 , we use the generalformula for Bessel functions of half-integer order from [16, p. 80, formula (12)]: K n + / ( z ) = (cid:16) π z (cid:17) / e − z n (cid:88) i =0 ( n + i )! i !( n − i )!(2 z ) i . Writing n − / = ( n −
2) + / , we see that G n ( (cid:126)r ) simplifies to G n ( (cid:126)r ) = 2 − n +1) e − r / π Γ( n ) n − (cid:88) i =0 ( i + ( n − i !(( n − − i )! r ( n − − i . In this form, it is clear that the only term in the sum which is nonzero at r = 0 is the i = n − term and G n ( (cid:126)
0) = 2 − n +1) (2 n − π Γ( n )( n − − n +1) Γ(2 n − π Γ( n )Γ( n − . We can simplify this a bit further using the duplication formula for gamma functions [6, 5.5.5]
Γ(2 z ) = π − / z − Γ( z )Γ( z + / ) to get C n = G n ( (cid:126)
0) = Γ( n − / )64 √ π Γ( n ) . (18)as desired.We can use the pdf for sums of edges in A ( n ) to write down the pdfs for single edges and pairsof edges in P ( n ) .9 Proposition 20.
The pdf of a single edge (cid:126)e i in P ( n ) with respect to dVol (cid:126)e i is spherically sym-metric on R and given by the following function of r i = | (cid:126)e i | : P ( (cid:126)e i ) = n − n − π Γ( n − / ) e − ri / r n − / i K n − / ( r i / ) . Proof.
The pdf of a single edge is just P ( (cid:126)e i ) = 1 C n g ( (cid:126)e i ) G n − ( − (cid:126)e i ); i.e., the probability of the i th edge being (cid:126)e i and the remaining n − edges summing to − (cid:126)e i ,conditioned on the assumption that all n edges sum to zero. Since g ( (cid:126)e i ) = G ( (cid:126)e i ) , we can useProposition 17 and Proposition 19 to arrive at the stated expression. Corollary 21.
The moments of edgelength for polygons in P ( n ) are E ( | (cid:126)e i | p ; P ( n ) , H) = ( n − n + p − n −
4) B( p + 2 , n − . and hence the expectation of squared chordlength and radius of gyration are E (Chord( k ); P ( n ) , H) = (cid:18) n − kn (cid:19) k (2 n − n + 1 E (Gyradius; P ( n ) , H) = (cid:18) n − n (cid:19) (2 n − . Proof.
Using the pdf from Proposition 20, the expected value of | (cid:126)e i | p = r pi is (cid:90) R r pi P ( (cid:126)e i ) dVol (cid:126)e i = (cid:90) ∞ (cid:90) π (cid:90) π r pi P ( (cid:126)e i ) r i sin φ d θ d φ d r i . Writing out P ( (cid:126)e i ) and integrating with respect to θ and φ gives the p th moment n − n − Γ( n − / ) (cid:90) ∞ e − ri / r n + p − / i K n − / ( r i / ) d r i = n − n − Γ( n − / ) √ π Γ( p + 2)Γ(2 n + p − n + p )= ( n − n + p − n −
4) B( p + 2 , n − using Lemma 16 and the duplication rule for the gamma function.0Since the measure on P ( n ) is invariant under rearrangement of edges, we can use Proposi-tions 6.3 and 6.5 from [2] as in Corollary 14. Specifically, E (Chord( k ); P ( n ) , H) = (cid:18) n − kn − (cid:19) kE ( | (cid:126)e i | ; P ( n ) , H) and E (Gyradius; P ( n ) , H) = (cid:18) n + 112 (cid:19) E ( | (cid:126)e i | ; P ( n ) , H) . By the first part of the proposition we have that E ( | (cid:126)e i | ; P ( n ) , H) = 12( n − n − n ( n + 1) , so the given formulas for E (Chord( k ); P ( n ) , H) and E (Gyradius; P ( n ) , H) are immediate. Proposition 22.
The probability distribution of a pair of edges (cid:126)e and (cid:126)e in P ( n ) is invariantunder the diagonal action of SO (3) on the pair of edges and invariant under rotations which fixone edge. Hence, it depends only on the lengths r and r of the two edges and the angle θ betweenthem. It is given by the formula P ( r , r , θ ) d r d r d θ = Γ( n )4 √ π Γ(2 n − r r e − ( r + r ) z n − K n − (cid:16) z (cid:17) sin θ d r d r d θ, (19) where z = | (cid:126)e + (cid:126)e | = (cid:112) r + r + 2 r r cos θ .Proof. As we saw above, the general form for the probability distribution of a pair of edges in aclosed polygon in P ( n ) is P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e = g ( (cid:126)e ) g ( (cid:126)e ) G n − ( − (cid:126)e − (cid:126)e ) C n dVol (cid:126)e dVol (cid:126)e , (20)that is, the probability of the first two edges being (cid:126)e and (cid:126)e and the remaining edges summingto − (cid:126)e − (cid:126)e conditioned on the assumption that all n edges sum to zero. We computed G k ( (cid:126)r ) inProposition 17 and C n in Proposition 19. Using the formula for g ( (cid:126)e ) from Proposition 9, we get P ( (cid:126)e , (cid:126)e ) dVol (cid:126)e dVol (cid:126)e = Γ( n )32 π / r r Γ(2 n − e − / ( r + r ) z n − / K n − / ( z / ) dVol (cid:126)e dVol (cid:126)e , where again z = | (cid:126)e + (cid:126)e | = (cid:112) r + r + 2 r r cos θ . We can rewrite (cid:126)e in spherical coordi-nates (cid:126)e = ( r , φ , θ ) . For each (cid:126)e , we can fix (cid:126)e as the z -axis of spherical coordinates for (cid:126)e = ( r , φ , θ ) . Here θ is equal to θ , the angle between (cid:126)e and (cid:126)e . Observing that we can inte-grate out φ and φ immediately to get a factor of π and θ to get a factor of , and recording thevolume form in these coordinates as r r sin θ d r d r d θ gives us the formula in the statement ofthe proposition.1 Corollary 23.
The pairwise distribution of edges (cid:126)e and (cid:126)e in P ( n ) for n > may also be writtenmore simply in terms of the variables x = ( r + r ) / , y = ( r − r ) / , and z = | (cid:126)e + (cid:126)e | . Inthese variables, the probability distribution is given by: P ( x, y, z ) d x d y d z = Γ( n )2 √ π Γ(2 n − e − x z n − / K n − / ( z / ) d x d y d z. (21) Proof.
Computing the Jacobian of the map ( r , r , θ ) (cid:55)→ ( x, y, z ) , we see that its inverse determi-nant is |J − | = 2 csc( θ ) (cid:112) r + r + 2 r r cos θr r We then multiply the function in Proposition 22 by this determinant and substitute to obtain thestatement of the Corollary.We now explicitly check that the total integral of the pairwise pdf is equal to 1. This computationwill serve as a warm-up for the more difficult definite integrals ahead.Since y does not appear in the pdf P ( x, y, z ) in (21), we will integrate with respect to y first.The triangle inequality states that r ≤ r + z , so y ≤ z / , and similarly r ≤ r + z , so − z / ≤ y .We will next integrate by x , since x does not appear in a Bessel function. There is no upper boundon x = ( r + r ) / , but since r + r ≥ z , we know x ≥ z / . We will integrate with respect to z last, and here the limits are simply and ∞ . Thus, we are trying to show that Γ( n )2 √ π Γ(2 n − (cid:90) ∞ (cid:90) ∞ z / (cid:90) z / − z / e − x z n − / K n − / ( z / ) d y d x d z = 1 . (22)The y and x integrals are simple, and leave us with Γ( n )2 √ π Γ(2 n − (cid:90) ∞ e − z / z n − / K n − / ( z / ) d z = 1 (23)by Lemma 16. This check gives us confidence that our pairwise pdf is correct so far. Proposition 24.
The expected value of the turning angle θ for a single pair of edges in P ( n ) isgiven by the formula E ( θ ) = π π n − (24)2 Proof.
Our overall strategy will be to use the formula for P ( x, y, z ) from Corollary 23 to write thisexpected value as E ( θ ) = (cid:90) ∞ (cid:90) ∞ z / (cid:90) z / − z / θ ( x, y, z ) P ( x, y, z ) d y d x d z. We begin by writing the turning angle θ in terms of the x , y , and z variables as θ ( x, y, z ) = arccos (cid:18) z − x + y )2( x − y ) (cid:19) . Since P ( x, y, z ) does not depend on y , the first integral is accomplished by integrating this functionwith respect to y . Integrating by parts with dv = 1 , we get (cid:90) z / − z / θ ( x, y, z ) d y = πz + (cid:90) z / − z / y √ x − z ( y − x ) (cid:112) z − y d y. Now, making the trig substitution y = z sin ψ yields πz + (cid:112) x − z (cid:90) π / − π / sin ψ sin ψ − x z d ψ = πz + (cid:112) x − z (cid:90) π / − π / (cid:18) − x x − z + z cos 2 ψ (cid:19) d ψ by using the identity sin ψ = − cos 2 ψ . This is simply πz + π (cid:112) x − z − πx using the indefinite integral identity [7, 2.558(4)] (cid:90) d ψa + b cos ψ = 2 √ a − b arctan ( a − b ) tan ψ / √ a − b . Since the function P ( x, y, z ) is in the form e − x f ( z ) , we must now do the pair of integrals π (cid:90) ∞ z e − x ( z − x ) d x + π (cid:90) ∞ z e − x (cid:112) x − z d x. The first integral is simple and has the value − πe − z / . To do the second integral, we make thechange of variables x = ( z / ) t to get (cid:90) ∞ z / e − x (cid:112) x − z d x = z (cid:90) ∞ e − t z ( t − d t = zK ( z / ) , K ν [6, 10.32.8] thatwe have used before. We have now shown that E ( θ ) = √ π Γ( n )2Γ(2 n − (cid:18)(cid:90) ∞ − e − z / z n − / K n − / ( z / ) d z + (cid:90) ∞ z n − / K ( z / ) K n − / ( z / ) d z (cid:19) = − n − n − π + √ π Γ( n )2Γ(2 n − (cid:90) ∞ z n − / K ( z / ) K n − / ( z / ) d z (25)where the integral is, as usual, computed using Lemma 16.The remaining integral in (25) is more interesting, as it involves a product of Bessel functions.Making the substitution z = 2 u , we can then use the Nicholson integral representation for theproduct of Bessel functions [6, 10.32.17] to rewrite the integral as √ π Γ( n )2Γ(2 n − (cid:90) ∞ z n − / K ( z / ) K n − / ( z / ) d z = √ π n − / Γ( n )Γ(2 n − (cid:90) ∞ cosh (( / − n ) t ) (cid:18)(cid:90) ∞ u n − / K n − / (2 u cosh t ) d u (cid:19) d t. Again, the inner integral is a power of z multiplied by a Bessel function of αz (here α = 2 cosh t )and can hence be evaluated using Lemma 16. We now have the integral π / − n ( n − n − (cid:90) ∞ cosh (( / − n ) t ) sech n − / t d t. (26)We will integrate this using the general integration formula (cid:90) ∞ cosh αx sech β x d x = (cid:114) π β + α )Γ( β − α ) P / − βα − / (0)Γ( β ) , (27)where P yx is the associated Legendre function. This is valid when β − α > and β + α > . It isa specialization of [7, 3.5.17]. We set α = / − n and β = n − / and see that β − α = 2 n − (which is positive since we have assumed n > ) and β + α = 4 . Applying the formula shows thatthe integral of (26) is equal to π / − n ( n − n − P − n − n (0)Γ(2 n − n − / ) = 4( n − n − n − n − π, (28)where we have used the general formula for the value at zero of associated Legendre functionsgiven in [6, 14.5.1] and the duplication formula for gamma functions to simplify the form on theleft hand side. Combining (25) and (28), we see that E ( θ ) = n − n − π = π π n − π π n − , (29)as desired.4Theorem 15 now follows by multiplying by n . We get an interesting corollary of this theoremfor n = 6 and n = 7 ; since the expected value of total curvature is less than π , some hexagonsand heptagons must have total curvature less than π and hence be unknotted by the F´ary-Milnortheorem [13]. Corollary 25.
If we measure volume using the symmetric measure on polygon space, at least / of the polygons in Pol (6) and / of the polygons in Pol (7) are unknotted.Proof. Let x be the fraction of polygons in Pol ( n ) with total curvature greater than π . By theF´ary-Milnor theorem, these are the only polygons which may be knotted. We know that any closedpolygon has total curvature at least π , so the expected value of total curvature satisfies E ( κ ; Pol ( n ) , σ ) > πx + 2 π (1 − x ) . Solving for x and using Theorem 15, we see that x < ( n − n − n − . For n > , this bound is not an improvement on the trivial bound x ≤ , but for n = 6 , we get x < / and for n = 7 , we get x < / , as desired.If instead we had let x be the fraction of polygons in Pol ( n ) with total curvature greater than πB , then x < ( n − n − B − n − and this gives constraints on the fraction of knots with bridge number B . For example, when n = 8 , , , or this gives us some information on the fraction of3-bridge knots since there are 3-bridge knots (e.g. ) with stick number 8.
5. NEW NUMERICAL METHODS FOR RANDOM POLYGONS
In [2], we gave a fast sampling algorithm for random polygons in
Pol ( n ) which is guaranteedto sample directly from the symmetric probability measure on this space given a supply of nor-mal random variates. The ensembles of polygons generated by this algorithm are as good as theunderlying ensembles of normals, so the quality of this sampling algorithm cannot be improved.However, when one is computing the expected value of a geometric functional on polygon spacesuch as total curvature, averaging over a large ensemble of sample polygons is simply Monte Carlointegration over a very high-dimensional space. The numerical accuracy of this method is neces-sarily limited.The framework above yields a much better method for computing expected values of functionslike total curvature which are both scale-invariant and sums of quantities that are defined locallyon a given polygon: integrate directly against the pdf for a finite collection of edges in a closed n -gon in P ( n ) . Carrying out this method is not trivial if the integrand, like turning angle, has5a singularity in most coordinate systems. However, the results are worth it. Figure 1 shows thecurvature “surplus” term of total curvature minus π n plotted with data from sampling and thenumber of correct digits obtained by sampling and by numerical integration. (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Number of edges n Total Curvature Surplus π π
10 15 20 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236)
Number of edges n Correct Digits
FIG. 1: On the left, we see the average total curvature surplus κ − π n plotted for ensembles of 5 millionpolygons with 5 to 25 edges generated by the direct sampling algorithm of [2] (dots), plotted together withthe exact formula for this surplus of π n n − given by Theorem 15 (curve) and the asymptotic value of thissurplus of π given by Theorem 1 (line). The fact that these three computations are in agreement servesas a useful check on our work above. On the right, we see the number of correct digits in the calculationof average total curvature by careful numerical integration of turning angle against the pairwise pdf ofCorollary 23 (top line, about 10 correct digits), averaging over ensembles of 5 million polygons (middleline, about 4 correct digits), and averaging over ensembles of 1 million polygons (bottom line, about 3correct digits). We can see that we obtain significantly better results by numerical integration.
6. FUTURE DIRECTIONS
It is clear that the methods above have many interesting applications. For instance, we can hopeto compute the expected total curvature for plane polygons as well. In this case Theorem 1 showsthat the expected curvature surplus is / π . However, the integral seems somewhat forbidding andwe do not have an explicit conjecture for a closed form.More promising is the direction of extending our results to other functionals on space polygons.Since Theorem 2 tells us that the expected value of any scale invariant functional over the space ofHopf-Gaussian polygons is equal to the expected value over polygons with the symmetric measure,we can expect to compute a number of other interesting expectations this way. For example, theargument of Section 2 can certainly be generalized to predict an asymptotic expected total torsionof π n − π . It would be interesting to find the exact expectation of total torsion. It seems possibleto apply these methods to average crossing number as well, which is certainly a topic for furtherinvestigation!6In Corollary 25, we used our expectation for total curvature with respect to the symmetric mea-sure to bound the fraction of unknotted fixed-length hexagons below by / and the fraction ofunknotted fixed-length heptagons below by / . Such bounds are clearly too small: numericallysampling ensembles of 5 million polygons shows the fraction of unknots to be roughly / , for hexagons and / for heptagons . Our bounds even significantly underestimate the frac-tion of fixed-length hexagons and heptagons with total curvature less than π , which a similarexperiment with 5 million samples reveals to be approximately . and , respectively. Wecould improve our bounds by computing the variance of total curvature, or even by finding anexplicit expression for the total curvature pdf. However, this will involve a more subtle globalanalysis of the correlations between turning angles in closed polygons, and we leave this topic forfuture work.
7. ACKNOWLEDGEMENTS
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