aa r X i v : . [ m a t h . M G ] A ug FILLING RANDOM CYCLES
FEDOR MANIN
Abstract.
We compute the asymptotic behavior of the average-case filling vol-ume for certain models of random Lipschitz cycles in the unit cube and sphere.For example, we estimate the minimal area of a Seifert surface for a model ofrandom knots first studied by Millett. This is a generalization of the classicalAjtai–Koml´os–Tusn´ady optimal matching theorem from combinatorial probabil-ity. The author hopes for applications to the topology of random links, randommaps between spheres, and other models of random geometric objects. Introduction
Main results.
This paper introduces a new kind of average-case isoperimetricinequality. Given a k -cycle Z on ([0 , n , ∂ [0 , n ), in any of a number of geometricmeasure theory senses, its filling volume F V ( Z ) is the minimal mass of a chain whoseboundary is Z . The well-known Federer–Fleming isoperimetric inequality [FF] statesthat for all k -cycles Z , F V ( Z ) ≤ C n,k mass( Z ) k +1 k and F V ( Z ) ≤ C n,k mass( Z ) . However, one might expect that most cycles of given mass are much easier to fill.Unfortunately, as explained to the author by Robert Young, a geometrically mean-ingful probability measure on the space of all cycles of mass ≤ N may be too muchto ask for. The issue is one of picking a scale: say we are trying to build a random1-Lipschitz curve in a finite-dimensional space. If the curve is to fluctuate randomlyat scale ε , then over time 1 it will only travel a distance on the order of √ ε . Thusthere is no way of ensuring random behavior in a scale-free way. This idea of decom-posing a finite-mass cycle into pieces at different scales can be made precise usingthe notion of a corona decomposition , as in [Jones] (in dimension 1) and [Young] (inhigher dimensions).On the other hand, there are a number of ways of, and a number of motivations for,building random “space-filling” cycles of mass O ( N ) which look essentially trivialon balls of radius N − /n . Our main theorems characterize three models of this formwhich exhibit similar isoperimetric behavior, and we hypothesize that this behaviorshould be generic for models whose randomness occurs mainly at large scales—anidea which may have a precise Fourier-analytic formulation.This isoperimetric behavior is described in codimension d by the functionAKT d ( N ) = √ N if d = 1 √ N log N if d = 2 N ( d − /d if d ≥ . The first inequality dominates when mass( Z ) <<
1, the second when mass( Z ) >> Theorem A.
Let Z be a k -cycle on S n obtained by sampling N oriented great k -spheres independently from the uniform distribution on the oriented Grassmannian f Gr k +1 ( R n +1 ) . Then there are constants C > c > depending on n and k such that (1.1) c AKT n − k ( N ) < E ( F V ( Z )) < C AKT n − k ( N ) . Moreover,
F V ( Z ) is concentrated around its mean: there are constants C , C > depending on n and k such that (1.2) for every r > , P [ | F V ( Z ) − E ( F V ( Z )) | ≥ r ] ≤ C exp( − C √ N r )From (1.2) we see that the spread of the distribution around the mean is at moston the order of √ N ; in codimensions d = n − k ≥
2, this is small compared to themean:for every ε > | F V ( Z ) − E ( F V ( Z )) | E ( F V ( Z )) ≤ ε with high probability as N → ∞ . If a model of random codimension- d cycles of mass O ( N ) satisfies (1.1) and (1.2),we say it exhibits AKT statistics , in honor of Ajtai, Koml´os, and Tusn´ady, whodiscovered this phenomenon in the setting k = 0.By rescaling the picture, we can make this result more interpretable. Let R = T / ( n − k ) . Then the corresponding process in the n -sphere of radius R generates acycle of mass Θ( R n ) which is evenly spread throughout the sphere, so that a 1-ballintersects one of the great k -spheres in Z on average. With that rescaling, the massof an optimal filling becomes Θ( R n √ R ) if k = n − R n √ log R ) if k = n − R n ) if k ≤ n − . Informally speaking, to meet its match, the average point in Z has to travel a dis-tance Θ( √ R ) (in codimension 1), Θ( √ log R ) (in codimension 2), or Θ(1) (otherwise)times the distance to its closest neighbor.We also prove similar results for the cube: Theorem B.
Let Z be a relative k -cycle on ([0 , n , ∂ [0 , n ) obtained by sampling N planes independently from the uniform distribution on the space Y of oriented k -planes which intersect [0 , n nontrivially. Then Z exhibits AKT statistics. There may be reasonable disagreement as to which distribution is the uniformone in this context; to prove (1.1) it suffices to require that it be uniform on eachsubset of Y (isometric to two copies of a polytope) consisting of parallel planes, butto prove (1.2) we also need to assume that it behaves reasonably with respect to themanifold structure on Y (for example, is a positive density or has finite support).In fact, the only thing used here about Z is that almost all of its “slices” alongcoordinate k -planes consist of O ( N ) independent uniformly distributed points. Thismeans that there are a number of other possible models that can be fit into thisframework. However, the following requires a separate proof: Theorem C.
Let { M N } be a sequence of k -dimensional oriented pseudomanifoldswith N vertices and at most L simplices incident to any given simplex. Let Z bea k -cycle on [0 , n obtained by sending each vertex of M N to a uniformly randompoint in [0 , n and extending linearly. Then Z exhibits AKT statistics. ILLING RANDOM CYCLES 3
In the context of this theorem, the constants in (1.1) depend on n , k , and L , butnot on the shapes of the pseudomanifolds (which can therefore also be randomized).The case k = 1, n = 3 describes the “random jump” model of random knots andlinks introduced by Millett [Mil]. Moreover, by a theorem of Hardt and Simon [HS],the optimal filling of such a knot or link (after a slight rounding of corners) is a C embedded surface. In particular: Corollary 1.3.
For some
C > c > , the minimal Seifert surface of a knot pro-duced using N random jumps has area between c √ N log N and C √ N log N withhigh probability. Motivation.
The methods we use to prove Theorems A, B, and C can beeasily extended to other i.i.d. samples of simple shapes on various spaces. However,the investigation is mainly motivated by the desire to analyze topological invariantsof random geometric objects such as links and maps. Models of such objects tend toproduce random cycles which are similarly trivial at small scales, but are otherwisemore difficult to work with.
Random knots and links.
There have been a number of proposed models of randomknots and links; see [E-Z] for a detailed survey. Several of these models are “spatial”in the sense that they produce random knotted curves in space, and one supposesthat these may exhibit AKT statistics for filling area. As mentioned above, weshow this for Millett’s random jump model, but it may also be true for randompolygonal walks with shorter segments as well as random grid walks, perhaps withsome restrictions on segment length.Given two random curves in a certain model, one may want to understand thedistribution of their linking number. Since this will usually be zero on average,the first interesting question is about the second moment. Linking number can becomputed as the intersection number of one curve with a filling of the other, thusone may expect that two random curves of length N which exhibit AKT statisticshave expected squared linking number ∼ N log N or ∼ N √ log N .However, this is not the case for the Millett model: the second moment of thelinking number between two random jump curves of length N is ∼ N [ABD + , FK].Similarly, one may take the setup of Theorem A for k = 1 and n = 3 as a model of arandom link and try to understand the total linking number , that is, the sum of thesigned linking numbers of all pairs of circles. This is then the intersection numberof the chain with its own filling. Here it is easy to see that the second moment ofthe distribution is once again ∼ N .In both cases, this seeming incongruity perhaps boils down once again to theissue of multiple scales: random jump curves and great circles only “see” the largestscales, but the lower bound on filling volume in codimension 2 comes from lookingon many different scales at once. One may perhaps get a different answer mosteasily by analyzing the linking number of an asymmetric model: a random jumpcurve and a random walk of total length N made of smaller segments.In [Tanaka,Marko], the second moment is computed for the linking number of tworandom walks; normalizing so that these walks have length N and expected diameter1, this second moment again becomes ∼ N . In this model, however, randomnesshappens at scale ∼ /N , so it is not expected to exhibit AKT statistics. FEDOR MANIN
Random maps.
Another way of producing a random (framed) link is as the preimageof a generic point under a random map f : S → S . In fact, the self-linking numberof this link is the Hopf invariant of the map, which is itself a natural subject forinvestigation since it is a complete topological invariant of such maps.One natural model of L -Lipschitz random maps is a uniformly random simplicial map from a triangulation of S at scale ∼ L to a tetrahedron. The maximal self-linking number of such a map is Θ( L ); on the other hand, the heuristics abovewould suggest that the second moment of the linking number of the random modelis between L and L log L .These ideas may have applications in topological and geometric data analysis,see [FKW].1.3. Methods.
The k = 0 cases of Theorems A and B are, up to minor adjustments,a classical theorem in combinatorial probability: Theorem 1.4 (Ajtai, Koml´os, and Tusn´ady [AKT]) . Let { X , . . . , X N } and { Y , . . . , Y N } be two sets of independent, uniformly distributed random points in [0 , d , and let L be the transportation cost between { X i } and { Y i } , that is, the total length ofan optimal matching. Then there are constants < c d < C d such that with highprobability, c d AKT d ( N ) < L < C d AKT d ( N ) . Since the original geometric proof in [AKT] of the most subtle case d = 2, thisand related results have been proved many other times, often by applying Fourieranalysis; see [BL19b] for further references and [Tal14] for a detailed treatment ofcertain analytic approaches. Another beautiful geometric proof of the upper boundon the sphere is due to Holden, Peres, and Zhai [HPZ].The proofs of Theorems A and B in general are obtained by applying the k = 0results to ( n − k )-dimensional slices of the cube and sphere. This is the reasonthat the results depend only on the codimension, and for the critical nature ofcodimension 2. The lower bound in (1.1) is obtained directly by integrating thelower bounds on these slices. The upper bound is obtained via a dual result ondifferential forms; this kind of technique was already used in [AKT] for the proofof the lower bound for the square. Finally, (1.2) is proved using the notion ofconcentration of measure due originally to Gromov and Milman [GM]; see [Led] foran extensive modern treatment.Theorem C is proved similarly, except that slices no longer consist of i.i.d. points.Even this small amount of dependence complicates the argument considerably. Weuse ad hoc combinatorial arguments to overcome this, but one might hope to gener-alize, for example by applying a variant of Stein’s method, to a version of Theorem1.4 in the presence of dependence (one approach, which only gives upper bounds, isdiscussed in [BL19b, § Structure of the paper.
Section 2 introduces necessary ideas and results fromgeometric measure theory, and Section 3 discusses the classical AKT theorem. InSections 4 and 5, the upper and lower bounds in Theorems A and B are provedusing tools that may generalize to other models of random cycles. In Section 6, wediscuss the extra ideas needed to prove Theorem C. Finally, Section 7 discusses theconcentration of the distributions in these theorems around their mean.
ILLING RANDOM CYCLES 5
Acknowledgements.
I would like to thank Matthew Kahle and Robert Young for alarge number of helpful discussions over a span of three years. Yevgeny Liokumovichprovided a crucial reference; Shmuel Weinberger asked a question which inspiredTheorem C and gave other helpful comments. I was partially supported by NSFindividual grant DMS-2001042.2.
Definitions and preliminaries
Cycles and currents.
There are a number of useful ways to define chains andcycles from the point of view of topology and geometric measure theory. Algebraictopology typically uses singular k -chains: formal linear combinations of continuousmaps from the k -simplex to a topological space X (“singular simplices”). We willusually restrict our attention to Lipschitz simplices (that is, requiring the maps tobe Lipschitz) on a Riemannian manifold M . By Rademacher’s theorem, a Lipschitzsimplex σ : ∆ k → M is differentiable almost everywhere and so has a well-definedvolume or mass , mass( σ ) = Z ∆ k σ ∗ d vol M . We can then extend by linearity to define the mass of a Lipschitz chain.A more general notion of chain is that of a normal current. A k -dimensional current on a manifold M is simply a functional on (smooth) differential forms,which we think of as integration over the current. For example: • Every Lipschitz chain T defines a current via ω R T ω . • Every compactly supported ( n − k )-form α ∈ Ω n − k ( M ) defines a current via ω R M α ∧ ω .The boundary operator is defined via Stokes’ theorem: for a current T , ∂T ( ω ) = T ( dω ) . The mass of a k -current T on M , which agrees with the same notion on Lipschitzchains, is defined to bemass( T ) = inf { T ( ω ) : ω ∈ Ω k ( M ) and k ω k ∞ = 1 } . Here k ω k ∞ is the supremum of the value of ω over all frames of unit vectors. Fora general current, the mass of course need not be finite. A current T is normal if T and ∂T both have finite mass; in particular any cycle (current with emptyboundary) of finite mass is normal.2.2. Fillings and duality.
Now, if S is a normal current such that ∂S = T , wecall it a filling of T . The filling volume of T is F V ( T ) = inf { mass( S ) | ∂S = T } , which is always finite by the work of Federer and Fleming. The following is aninstance of the Hahn–Banach theorem: Proposition 2.1.
Let M be a manifold. Then a normal k -current T in M with ∂T = 0 has a filling of mass c if and only if for every ω ∈ Ω k ( M ) with k dω k ∞ ≤ , R T ω ≤ c .More generally, for any closed set A ⊂ M , write Ω k ( M, A ) for the vector space offorms whose restriction to A is zero. Let T be a normal k -current with ∂T supported FEDOR MANIN on A , that is, such that R ∂T α = 0 for any ( k − -form α ∈ Ω k − ( M, A ) . Then T has a filling relative to A (that is, a ( k + 1) -current S such that ∂S − T is supportedon A ) of mass c if and only if for every ω ∈ Ω k ( M, A ) with k dω k ∞ ≤ , R T ω ≤ c . In other words, the filling volume of a cycle T can be redefined as F V ( T ) = sup (cid:8)R T ω | ω ∈ Ω k ( M ) such that k dω k ∞ ≤ (cid:9) in both the absolute and the relative case. Our proofs of the upper bounds inTheorems A and B will be based on this proposition rather than constructing fillingsdirectly.Of course, knowing that a nice Lipschitz cycle has a filling which is a normalcurrent is not very satisfying—after all, normal currents can still be very strange.Luckily, given a normal current filling, we can upgrade it to a Lipschitz chain (at thecost of multiplying the mass by a constant) using the following classical theorem: Theorem 2.2 (Federer–Fleming deformation theorem [FF, Thm. 5.5]) . There is aconstant ρ ( k, n ) = 2 n k +2 such that the following holds. Let T be a normal currentin N k ( R n ) . Then for every ε > we can write T = P + Q + ∂S , where(1) mass( P ) ≤ ρ ( k, n ) mass( T ) .(2) mass( Q ) ≤ ερ ( k, n ) mass( ∂T ) .(3) mass( S ) ≤ ερ ( k, n ) mass( T ) .(4) P is a polyhedral cycle which can be expressed as an R -linear combinationof k -cells in the cubical unit lattice in R n .(5) If T is a Lipschitz chain, then so are Q and S .(6) If ∂T is a Lipschitz chain, then so is Q . If T is a normal current filling a Lipschitz chain ∂T , then P + Q is a Lipschitzchain filling T whose mass is only greater by a multiplicative constant ρ ( k, n ).It is not hard to upgrade the deformation theorem to manifolds, although theresulting constants will depend on the manifold and its metric; see for example[EPC + , Theorem 10.3.3].2.3. Slicing.
An important property of normal currents, introduced in [FF, § § T be a k -current on a manifold M . Given a Lipschitz function u : M → R ,for almost all r ∈ R there is a ( k − T ∩ { u ( x ) = r } with the followingproperties:(1) If T is defined by integration over a (rectifiable) set X ⊂ M , then T ∩{ u ( x ) = r } is defined by integration over M ∩ { u ( x ) = r } .(2) ∂T ∩ { u ( x ) = r } = ∂ ( T ∩ { u ( x ) = r } ).(3) mass( T ) ≥ u R ∞−∞ mass( T ∩ { u ( x ) = r } ) dr. In particular, by inductively slicing in different directions, we get the following:
Proposition 2.3.
Let ≤ k ≤ m ≤ n , and let T be an m -dimensional current on [0 , n . Given ~x = ( x , . . . , x k ) ∈ R k , let P ~x be the plane { ~x } × R n − k . Then for ILLING RANDOM CYCLES 7 almost all ~x , there is a well-defined ( m − k ) -dimensional current T ∩ P ~x such that ∂ ( T ∩ P ~x ) = ∂T ∩ P ~x mass( T ) ≥ Z [0 , k mass( T ∩ P ~x ) d~x. A variation on the Ajtai–Koml´os–Tusn´ady theorem
The results of this paper are a generalization of Theorem 1.4. Properly, thetheorem of Ajtai, Koml´os, and Tusn´ady [AKT] is in the case n = 2; their paperalso asserts the case n ≥
3, which is easy and later proved and extended in severaldirections by Talagrand [Tal92, Tal94]. The n = 1 case is elementary, and the proofalong with a vast array of strengthenings and generalizations can be found in [BL19a]by Bobkov and Ledoux. Here we need a slight variation. Theorem 3.1.
Generate a cycle Z of mass N in C ([0 , n , ∂ [0 , n ) by selecting N independent, uniformly distributed points in [0 , n × { +1 , − } . Then for there areconstants < c n < C n such that c n AKT n ( N ) ≤ E ( F V ( Z )) ≤ C n AKT n ( N ) . Remark 3.2.
Suppose that D is a Riemannian ball diffeomorphic to [0 , n andhas a volume form. Then by the main theorem of [BMPR] (extending results ofMoser [Moser] and Banyaga [Bany]), there is a diffeomorphism between the twowhich multiplies the volume form by a constant. Therefore Theorem 3.1 also holdswith respect to Lebesgue measure on D , with constants 0 < c D < C D depending onthe ratio of the volumes and the bilipschitz constant of this diffeomorphism.Moreover, given a smooth family of Riemannian balls, [BMPR] indicates thatthere is a smooth family of such diffeomorphisms. Therefore, if the family is com-pact, one can find uniform constants for the whole family. Proof.
There are two differences here from the results as they are typically presentedin the probability literature, where the problem consists of matching two sets ofrandom points of the same cardinality: first, the number of positive and negativepoints may not match; second, we are allowed to match points to the boundary aswell as to points of the opposite orientation. We briefly explain how to modify theoriginal proofs to deal with this.Clearly, the possibility of matching to the boundary cannot make the upperbounds worse. Let’s say without loss of generality there are more positive points.To obtain the upper bound for n ≥
2, we may simply ignore some arbitrary set of“extra” positive points, matching all the others first. By the central limit theorem,the expected number of extra points is O ( √ N ), so the extra mass generated bymatching them all to the boundary of the cube does not change the asymptoticanswer.For the lower bound in the case n = 2, we use the same stratagem of ignoringthe “extra” points to create a new cycle Z ′ . From the original proof in [AKT], weknow that there is a 1-Lipschitz function f : [0 , → R which is zero on ∂ [0 , and such that R Z ′ f ≥ c √ N log N with high probability. Since with high probability In fact, a somewhat similar, but more complicated modification was studied by Shor [Shor].
FEDOR MANIN the number of extra points is << √ N log N , and the values of f lie between − / /
2, we also know that R Z f ≥ c √ N log N with high probability.The lower bound in the case n ≥ ≥ cN − /n from everypositive point and the boundary, where c > n .In the case n = 1, the filling is unique up to a constant: the unique filling F supported away from zero has density R x Z at x ∈ [0 , §
3] to give estimates on E (mass F ).The upper bound is a simple calculation: E (mass F ) = Z E (cid:0)(cid:12)(cid:12)R x Z (cid:12)(cid:12)(cid:1) dx ≤ Z q Var( R x Z ) dx = √ N . The lower bound comes from the following classical fact, found in [BL19a] as Lemma3.4:
Lemma 3.3.
Given independent mean zero random variables ξ , . . . , ξ N , E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 ξ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≥ √ E (cid:18)(cid:18) N X k =1 ξ k (cid:19) / (cid:19) . Let ( X k , σ k ) ∈ [0 , × { +1 , − } be the k th chosen point. Then applying thelemma to ξ k = σ k χ { X k ≤ x } , we get E (cid:0)(cid:12)(cid:12)R x Z (cid:12)(cid:12)(cid:1) ≥ √ E (cid:18)(cid:18) N X k =1 ξ k (cid:19) / (cid:19) ≥ √ (cid:18) N X k =1 ( E ( | ξ k | )) (cid:19) / = 12 √ √ N x, and therefore E (mass F ) ≥ p N/ (cid:3) Proof of the upper bound
To prove the upper bound in Theorems A and B, we will use Stokes’ theorem;that is, we use the fact that for a cycle Z ∈ C k ( M, A ),(4.1)
F V ( Z ) = sup (cid:8)R Z α : α ∈ Ω k ( M, A ) such that k dα k ∞ = 1 (cid:9) . In fact, since Z is a cycle, R Z α only depends on ω = dα . To bound this quantity,we first note that any ω ∈ Ω k +1 ([0 , n ) can be decomposed into a sum of “basic”forms of the form ω I ( x ) dx i ∧ · · · ∧ dx i k +1 , where ω I is a function R n → R , for each subset { i , . . . , i k +1 } ⊂ { , . . . , n } . Lemma 4.2.
For any exact form ω ∈ Ω k +1 ([0 , n , ∂ [0 , n ) (resp., ω ∈ Ω k +1 ([0 , n ) ),there is a form α ∈ Ω k ([0 , n , ∂ [0 , n ) (resp., α ∈ Ω k ([0 , n ) ) given by α = X I ⊂ [ n ] | I | = k α I ( x ) dx I , ILLING RANDOM CYCLES 9 such that dα = ω , and for each I , k α I k Lip ≤ C n,k k ω k ∞ .Proof. We prove this by induction on n and k , keeping n − k constant. The case k = 0 is clear, since then ω is the zero function.To do the inductive step in the relative case, we follow the usual proof of thePoincar´e lemma with compact support, following [BottTu, § ε : [0 , → [0 ,
1] which is 0 near 0 and 1 near 1. By applyingthe lemma one dimension lower, we get a form η ∈ Ω k − ([0 , n − , ∂ [0 , n − ) with dη = R ω and k η I k Lip ≤ C n − ,k − k ω k ∞ . Then ω = dα for α = R t ω − ε ( x n ) π ∗ ( R ω ) − dε ( x n ) ∧ π ∗ η, where π is the projection to the ( n − x n . Notice that α I = − dεdx η I \{ n } if n ∈ I R t ω I ∪{ n } − ε ( x ) π ∗ ( R ω I ∪{ n } ) otherwise.This gives us a bound on the k α I k Lip in terms of the k ω I k Lip and k η I k Lip as well asthe derivatives of ε .For the non-relative version, we follow the same proof, mutatis mutandis, taking α = R t ω − π ∗ η. (cid:3) Theorem 4.3.
Let Z be a random k -cycle in ([0 , n , ∂ [0 , n ) or in [0 , n such thatfor some M > , (4.4) E ( F V ( Z ∩ P )) ≤ M for almost all coordinate ( n − k ) -planes P (in particular, Z ∩ P is almost always awell-defined zero-cycle). Then (4.5) E ( F V ( Z )) ≤ (cid:18) nk (cid:19) C n,k M, where C n,k is the constant from Lemma 4.2.Proof. For a k -form α , R Z α depends only on dα . Therefore to estimate (4.1) it isenough to show that for any ( k + 1)-form ω with k ω k ∞ = 1, there is a k -form α such that dα = ω and R Z α ≤ C n,k M .By Lemma 4.2, we can choose α = X I ⊂ [ n ] | I | = k α I ( x ) dx I such that k dα I k ∞ ≤ C n,k for every I . Then for α ranging over all these choices ofantidifferentials, F V ( Z ) = sup α Z Z α = sup α X I ⊂ [ n ] | I | = k Z R Ic Z Z ∩ P u α I du ≤ X I ⊂ [ n ] | I | = k Z R Ic (cid:18) sup α Z Z ∩ P u α I (cid:19) du ≤ X I ⊂ [ n ] | I | = k Z R Ic C n,k F V ( Z ∩ P u ) . Therefore E ( F V ( Z )) ≤ X I ⊂ [ n ] | I | = k Z R Ic C n,k E ( F V ( Z ∩ P u )) ≤ (cid:18) nk (cid:19) C n,k M. (cid:3) Proof of Theorem B, upper bound.
Let Z be a cycle in C k ([0 , n , ∂ [0 , n ) obtainedby sampling N i.i.d. planes from a distribution on the set of oriented k -planes whichintersect nontrivially with [0 , n , such that the distribution is uniform (with respectto Lebesgue measure on the corresponding polytope in R n − k ) on each set of parallelplanes.This condition clearly implies that for every coordinate ( n − k )-plane P , Z ∩ P consists of at most N i.i.d. positive and negative points with probability 1. ThenTheorem 3.1 implies that (4.4) holds for Z with M = C n − k AKT n − k ( N ). E ( F V ( Z )) ≤ (cid:18) nk (cid:19) C n,k C n AKT n − k ( N ) . (cid:3) Proof of Theorem A, upper bound.
We use the fact that the transverse intersectionof an oriented great k -sphere with an oriented great ( n − k )-sphere is a pair ofantipodal points with opposite orientations. Therefore, if Z is a cycle obtained bysampling N oriented great k -spheres independently from the uniform distribution,then for any great ( n − k )-sphere P , with probability 1 Z ∩ P = N X i =1 [ x i ] − [ − x i ]where the x i are i.i.d. uniform points on S n − k .Consider S n as a subset of R n +1 , with standard unit basis vectors e , . . . , e n .Let K ± i be the preimage of the cube [ − R, R ] n under central projection (that is,projection along lines through the origin) to the plane x i = ±
1. If R is largeenough, the interiors of the K ± i cover S n . Each K + i is disjoint from its antipodalset K − i ; therefore for any great ( n − k )-sphere P , with probability 1 Z ∩ P ∩ K ± i consists of i.i.d. uniform points. By Remark 3.2, E ( F V ( Z ∩ P ∩ K ± i )) ≤ C n,k AKT n − k ( N ) , where Z ∩ P ∩ K ± i is considered as a cycle in C ( P ∩ K ± i , ∂ ( P ∩ K ± i )).Note that central projection sends great spheres to hyperplanes. Therefore, byTheorem 4.3, for each i and sign,(4.6) E ( F V ( Z ∩ K ± i )) ≤ C n,k AKT n − k ( N ) . Set a partition of unity { ϕ ± i } subordinate to { K ± i } which is invariant with respectto the involution, that is, such that ϕ + i ( x ) = ϕ − i ( − x ). To prove the theorem, it isenough, given a k -form ω ∈ C k +1 ( S n ) with k ω k ∞ = 1, to show that for some (andtherefore every) α ∈ C k ( S n ) with dα = ω , Z Z α ≤ C n,k AKT n − k ( N ) . ILLING RANDOM CYCLES 11
But note that Z Z α = n X i =0 (cid:16)Z Z ∩ K + i ϕ + i α + Z Z ∩ K − i ϕ − i α (cid:17) . Therefore it suffices to find an antidifferential and a bound separately for each ϕ ± i ω .Therefore (4.6) suffices to prove the theorem. (cid:3) Proof of the lower bound
Theorem 5.1.
Let Z be a random Lipschitz k -cycle in ([0 , n , ∂ [0 , n ) such thatfor almost every k -plane P ~x = { ( x , . . . , x k ) } × [0 , n − k ⊂ [0 , n , the slice Z ∩ P ~x satisfies E ( F V ( Z ∩ P ~x ) ≥ p ( ~x ) where p : [0 , k → [0 , ∞ ) is an L function. Then E ( F V ( Z )) ≥ Z [0 , k p ( ~x ) d~x. Proof.
Let U be a normal current filling Z such that mass( U ) ≤ F V ( Z ) + ε , for any ε >
0. Then for almost all P ~x , there is a slice U ∩ P ~x which fills Z ∩ P ~x , andmass( U ∩ P ~x ) ≥ F V ( Z ∩ P ~x ) . By Prop. 2.3,
F V ( Z ) + ε ≥ mass( U ) ≥ Z [0 , k mass( U ∩ P ~x ) d~x ≥ Z [0 , k p ( ~x ) d~x. Since this is true for every ε >
0, the proof is complete. (cid:3)
Proof of Theorem B, lower bound.
Let Z be a cycle in C k ([0 , n , ∂ [0 , n ) obtainedby sampling N i.i.d. planes from a distribution on the set of oriented k -planes whichintersect nontrivially with [0 , n , such that the distribution is uniform (with respectto Lebesgue measure on the corresponding polytope in R n − k ) on each set of parallelplanes. Assume, perhaps by permuting coordinates, that this distribution is notconcentrated on planes of the form P ~x = ( x , . . . , x k ) × R n − k . As in the proof of the upper bound, it follows that for every P ~x , Z ∩ P ~x consistsof i.i.d. positive and negative points with probability 1. Moreover, the probabilityof a random plane P intersecting P ~x inside [0 , n depends only on the direction of P and not on ~x . Thus E (mass( Z ∩ P ~x )) ≥ cN, where c depends on the distribution but not on ~x .Thus by Theorems 3.1 and 5.1, E ( F V ( Z )) ≥ C n − k AKT n − k ( cN ) . (cid:3) Proof of Theorem A, lower bound.
Let Z be a cycle in C k ( S n ) obtained by sampling N independent uniformly distributed great k -spheres. It suffices to show that forsome compact submanifold K ⊂ S n , F V ( Z ∩ K ) ≥ C n,k AKT n − k ( N )where Z ∩ K is considered as a cycle in C k ( K, ∂K ).Recall that for any great ( n − k )-sphere P , with probability 1 Z ∩ P = N X i =1 [ x i ] − [ − x i ]where the x i are i.i.d. uniform points on S n − k . Let T ⊂ S n be a great k -sphere and T ′ the ( n − k )-sphere consisting of points farthest from T . We use N ε ( U ) to indicatethe ε -neighborhood of the set U ; then S n \ N π/ ( T ′ ) = N π/ ( T ) deformation retractsto T along the orthogonal retraction ρ : N π/ ( T ) → T . We let K = ρ − ( K ′ ) where K ′ is some closed ball in T which does not include any point and its antipode.Notice that K is foliated by equal-volume patches of great ( n − k )-spheres P u whichretract to points u ∈ K ′ , and also does not include any point and its antipode. ByRemark 3.2, there is a bilipschitz diffeomorphism from K to [0 , n which sends each P u to a plane of the form ( x , . . . , x k × R n − k in a volume-preserving way (up to a constant). Therefore, for each u ∈ K ′ , E ( F V ( Z ∩ P u ∩ K )) ≥ c AKT n − k ( cN ) , and applying Theorem 5.1, we obtain the result. (cid:3) Proof of Theorem C
Before we prove Theorem C, we must give a more precise statement.By an oriented k -pseudomanifold we mean a k -dimensional simplicial complex M with the following properties: • It is pure , i.e. every simplex is contained in an k -dimensional simplex. • Every k -simplex comes with an orientation such that the sum of all theoriented k -simplices is a cycle in C k ( M ).Note that this is considerably wider than the usual definition: it is just enough sothat if M is equipped with the standard simplexwise metric, any Lipschitz map from M to a metric space X defines a Lipschitz k -cycle in X .We say M has geometry bounded by L if every k -simplex intersects at most L others.With these definitions, we restate Theorem C: Theorem.
Let M be an oriented k -pseudomanifold with N vertices and geometrybounded by L . Let Z be a k -cycle on [0 , n obtained by sending each vertex of M toa uniformly random point in [0 , n and extending linearly. Then there are constants C > c > depending on n and k such that (6.1) cL − AKT n − k ( N ) < E ( F V ( Z )) < CL AKT n − k ( N ) . ILLING RANDOM CYCLES 13
The concentration result will be proved in the next section.As with Theorem B, the proof is a direct application of Theorems 4.3 and 5.1. Toapply these theorems, we need to understand the filling volumes of slices of Z , whichis more complicated in this case because while the points are identically distributed,they are not entirely independent. This is achieved for the upper bound in Lemma6.3 and for the lower bound in Theorem 6.12. Together, these complete the proof.In this section as before, fix the notation P ~x = { ~x } × [0 , n − k ⊂ [0 , n , ~x ∈ [0 , k . We start by analyzing the slice Z ∩ P ~x . Lemma 6.2.
Let ~x ∈ [0 , k . Then the slice Z ∩ P ~x is the sum of N random -chains ζ , . . . , ζ N which are identically distributed on {± [ y ] : y ∈ [0 , n − k } ∪ { } accordingto a distribution µ ~x depending on k and ~x . Moreover:(i) µ ~x is invariant with respect to the involution sending a chain ζ to − ζ ;(ii) µ ~x ≤ C ( n, k ) µ Lebesgue on [0 , n − k .(iii) Each ζ i is independent of all but at most L other ζ j . Since the distribution of Z is invariant under permuting coordinates, this holdsfor any ( n − k )-dimensional slice in a coordinate direction. Proof.
The distribution in question is the intersection of a random linear k -simplexin [0 , n with P ~x . Property (i) is obvious from this, and (iii) follows since a pair of ζ i are independent whenever the two corresponding simplices do not intersect. Tosee (ii), consider the function F : (cid:0) ([0 , n ) k +1 , µ Lebesgue (cid:1) → (cid:0) {± [ y ] : y ∈ [0 , n − k } ∪ { } , µ ~x (cid:1) sending each linear k -simplex to its intersection with P ~x . This function is measure-preserving by definition, and its restriction to K = F − { [ y ] : y ∈ [0 , n − k } is 1-Lipschitz. Therefore, by the coarea formula, the density function of µ ~x is givenby the [ n ( k + 1) − ( n − k )]-dimensional Hausdorff measure of point preimages. Thusit is enough to bound H ( n +1) k ( F − ( ~y )) for each ~y .Let T be the set of linear k -simplices with vertices in [ − , n which pass through ~
0, and let T ~x = ( T + ( ~x,~ ∩ ([0 , k × [ − , n − k ) k +1 . (Here each vertex is translated by ( ~x,~ F − ( ~y ) ⊂ T ~x + ( ~ , ~y ) . All these translates are disjoint and their union is a subset of ([0 , k × [ − , n − k ) k .Therefore, again by the coarea formula, H ( n +1) k ( T ~x ) ≤ ( k +1)( n − k ) . This completes the proof of (ii). (cid:3)
Condition (iii) gives a dependency graph of degree ≤ L between the ζ i . Thisgraph has an ( L + 1)-coloring, giving a partition of { , . . . , N } into L + 1 disjointsubsets I , . . . , I n such that for i ∈ I j , the ζ i are i.i.d.The following lemma gives the upper bound to plug into Theorem 4.3. Lemma 6.3.
For every ~x ∈ [0 , k , E ( F V ( Z ∩ P ~x )) ≤ ( L + 1) C n,k AKT n − k ( N ) . Proof.
Write ν ~x for the probability measure on [0 , n − k given by ν ~x ( A ) = µ ~x { [ ~y ] : ~y ∈ A } µ ~x { [ ~y ] : ~y ∈ [0 , n − k } . We first note that the upper bound from the original AKT theorem holds foreach ν ~x : if ζ is a random 0-cycle in [0 , n − k with N positive and N negative pointsdistributed according to ν ~x , then by [BL19b, equation (12)], for some constant C n − k independent of the measure,(6.4) E ( F V ( ζ )) ≤ C n − k AKT n − k ( N ) . To reduce to this situation, we note that while Z ∩ P ~x is a cycle (and therefore hasan equal number of negative and positive points), Z ( ~x, j ) = X i ∈ I j ζ i may not be. We produce cycles ˜ Z ( ~x, j ) for j = 1 , . . . , L + 1 by adding up to N additional i.i.d. points distributed according to ν ~x . We add each point to ˜ Z ( ~x, j ) fortwo different j , with opposite signs, so that L +1 X j =1 ˜ Z ( ~x, j ) = L +1 X j =1 Z ( ~x, j ) = Z ∩ P ~x . Each ˜ Z ( ~x, j ) is a 0-cycle consisting of at most N positive and N negative i.i.d. points.Therefore, by (6.4), E ( F V ( Z ∩ P ~x )) ≤ ( L + 1) C n,k AKT n − k ( N ) . (cid:3) The lower bound is somewhat more difficult and forces us to go back to theoriginal proof of the AKT theorem in various dimensions. We start by estimatingthe relationship between correlated points in Z ∩ P ~x . For this we need to betterunderstand the function F and set T defined in the proof of Lemma 6.2. Lemma 6.5.
Given a linear k -simplex ∆ ∈ ([0 , n ) k +1 such that at least one of its ( k − -faces is at distance at least r from P ~x , q det (cid:0) ( DF ∆ ) T DF ∆ (cid:1) ≥ C n,k r n − k . Proof.
From Figure 1, we see that for every 1 ≤ i ≤ n − k , there is a unit vector ~v such that DF ∆ ( ~v ) = c~e i , for c > r/ √ k . Therefore q det (cid:0) ( DF ∆ ) T DF ∆ (cid:1) ≥ (cid:18) r √ k (cid:19) n − k . (cid:3) For the next lemma, we recall some facts about the set T . First, T ~x + (0 , y ) and T ~x + (0 , y ′ ) are disjoint if y = y ′ , and T ~x + { } × [0 , n − k ⊆ ([0 , k × [ − , n − k ) k +1 . ILLING RANDOM CYCLES 15 P ~x ∆ y i D ≤ √ k d ≥ r Figure 1.
When a vertex moves by ∆ y i , ∆ ∩ P ~x moves by ( d/D )∆ y i .(Here the notation T ~x + V refers to the translates of every simplex in T ~x by everypoint in V , in other words the Minkowski sum of T ~x with the diagonal { ( y, . . . , y ) : y ∈ V } ⊂ R n ( k +1) . )In fact, this containment still holds if we intersect both sides with U × [ − , ( n − k )( k +1) ,where U ⊆ [0 , k ( k +1) is any subset. Lemma 6.6.
Let Q ⊆ [1 / , / n − k be an open set, and suppose ~x ∈ [1 / , / k .Given a linear k -simplex ∆ with vertices in [0 , n , let ρ (∆) be the distance from theplane spanned by the first k vertices to { } × [0 , n − k . Then P (cid:2) ρ (∆) ≤ r | F (∆) = [ y ] , y ∈ Q (cid:3) ≤ C n,k r. Proof.
Note first that F − { [ y ] : y ∈ Q } ⊆ T ~x + Q , and in particular { ∆ : ρ (∆) ≤ r and F (∆) = [ y ] , y ∈ Q } ⊆ { ∆ ∈ T ~x : ρ (∆) ≤ r } + Q. Choosing k points randomly induces a probability measure on the set of ( k − R k whose density at a plane P is proportional to vol( P ∩ [0 , k ) k . Thisdensity is bounded above by some C n,k , and so the set of planes whose distancefrom ~x is at most r has measure ≤ C n,k r . Therefore P (cid:2) ρ (∆) ≤ r and F (∆) = [ y ] , y ∈ Q (cid:3) ≤ ( k +1)( n − k ) C n,k r vol Q. Thus it suffices to find a lower bound on vol( F − { [ y ] : y ∈ Q } ) in terms of vol Q .Here we use the restrictions on Q and ~x : clearly, F − { [ y ] : y ∈ Q } ⊇ [ T ~x ∩ ([ − / , / n + { ( ~x,~ } )] + Q. The volume of this set is independent of ~x , depends linearly on vol Q , and is easilyseen to be positive. (cid:3) Lemma 6.7.
Assume that ~x ∈ [1 / , / k . Let < r < and let ζ and ζ ′ be randomchains which encode the intersection with P ~x of two intersecting k -simplices ∆ and ∆ ′ of M . Let Q ⊂ [1 / , / n − k be a cube of side length ℓ . Then (6.8) P (cid:2) ζ ′ = ± [ y ′ ] , y ′ ∈ Q | ζ = [ y ] , y ∈ Q (cid:3) ≤ C n,k √ ℓ. Remark 6.9.
A more careful analysis based on the same principle shows that P (cid:2) ζ ′ = ± [ y ′ ] , y ′ ∈ Q | ζ = [ y ] , y ∈ Q (cid:3) ≤ ( C n,k ℓ | log ℓ | if n − k = 1 ,C n,k ℓ otherwise . Proof.
We give the proof in the case that ∆ and ∆ ′ share a ( k − ; thegeneral case is similar. Order the vertices so that the last is not shared between thetwo simplices, and call the images of those vertices w and w ′ ∈ [0 , n .Suppose first that n − k ≥
2. By Lemma 6.6,(6.10) P (cid:2) ρ (∆ ) ≤ √ ℓ | ζ = [ y ] , y ∈ Q (cid:3) ≤ C n,k √ ℓ. Now fix ∆ with ρ (∆ ) ≥ √ ℓ , and let U (∆ , Q ) ⊆ [0 , n be the set of points w ′ such that ζ ′ = ± [ y ′ ] with y ′ ∈ Q . Note that U (∆ , { z } ) is contained in a k -planeand hence its k -dimensional Hausdorff norm is at most some C n,k . So by Lemma6.5 and the coarea formula, (cid:18) ℓk (cid:19) n − k vol( U (∆ , Q )) ≤ C n,k vol( Q )and therefore vol( U (∆ , Q )) ≤ C n,k ℓ n − k . Integrating this over all possible values of ∆ , we see that(6.11) P (cid:2) ζ ′ = ± [ y ′ ] , y ′ ∈ Q | ζ = [ y ] , y ∈ Q, ρ (∆ ) ≥ √ ℓ (cid:3) ≤ C n,k ℓ n − k . Together, (6.10) and (6.11) imply (6.8). (cid:3)
Finally we have the tools we need to prove the lower bound for Theorem C.
Theorem 6.12.
For every ~x ∈ [1 / , / k , E ( F V ( Z ∩ P ~x )) ≥ c n,k L − AKT n − k ( N ) . Proof.
Write d = n − k . The arguments for d = 1, d = 2, and d ≥ ζ i be the chain-valued random variables corresponding to intersections of k -simplices of Z with P ~x . Recall that ζ i = ± [ v i ] or { } , with v i ∈ [0 , d identicallydistributed according to a density which is bounded below on [1 / , / d . Moreover,this bound is uniform with respect to ~x ∈ [1 / , / k . Thus, following Remark 3.2,there is a uniformly bilipschitz family of diffeomorphisms ϕ ~x : [1 / , / d → [0 , d which send this density to a constant times the standard volume form. In particular,Lemma 6.7 still holds for the ζ i after applying the diffeomorphism. We now write ζ i for ϕ ~x ( ζ i ).In each case, consider an ( L + 1)-coloring I , . . . , I L +1 of the dependency graphbetween the ζ i , with the colored subsets ordered from largest to smallest. Note thateach of the ζ i is correlated with ζ i ′ for at most L values of i ′ .We write Z j = P i ∈ I j ζ i . The rough plan is as follows: in each case, we show that Z is hard to fill by constructing a Lipschitz function f such that Z Z f ≥ c n,k L AKT n − k ( N ) Lip f with high probability.Then we show that E ( R Z j f ) for each j = 1 is very close to zero.In each case, the function f is constructed as a sum of simple functions. Given acube Q ⊆ [0 , d , let ∆ Q : [0 , d → R be the function supported on Q whose graph ILLING RANDOM CYCLES 17 is a symmetric pyramid with base Q and height 1. We also write ∆ rv for the cube inthe lattice of side length r which contains v ∈ [0 , d . The function f will consist of asum of scaled copies of ∆ Q , each reflecting the “imbalance” of positive and negativepoints on the cube Q . The main difference between different dimensions is the scaleof these cubes: for d ≥
3, the cubes are at the smallest scale (comparable to theaverage distance between neighboring points), for d = 1 they are at the largest scale(comparable to 1), and for d = 2 we use cubes at many scales, as in the originalproof of [AKT].In each case, we construct f by means of an auxiliary function g ( x ) = X i ∈ I g ζ i ( x )(in the case d = 2, each ζ i is associated to many summands g rζ i at different scales,which we consider separately). This g may not satisfy the desired upper bound onthe Lipschitz constant, so we remove some of the summands where they are tooconcentrated to produce f . Whenever ζ i is independent from ζ i ′ , E (cid:0)R ζ i ′ g ζ i ( x ) (cid:1) = 0,and so for every j = 1 we can write (cid:12)(cid:12) E (cid:0)R Z j g (cid:1)(cid:12)(cid:12) ≤ X i ∈ I j X { (cid:12)(cid:12) E (cid:0)R ζ i g ζ i ′ (cid:1)(cid:12)(cid:12) | ζ i ′ is correlated with ζ i } . By Lemma 6.7, this correlation is not too high, and therefore we can bound each ofthe summands. By the construction of f , this also bounds (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) .If we tune everything correctly, we get that for each j = 1, (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) ≤ L E (cid:0)R Z f (cid:1) , giving a lower bound on E (cid:0)R Z f (cid:1) . Case d ≥ . We split the cube [0 , d into ∼ N subcubes of side length r ≈ N − /d ,and let g ( x ) = X i ∈ I g i ( x ) = X i ∈ I ± r ∆ rv i ( x ) where ζ i = ± [ v i ] ,f ( x ) = r − − X t ,...,t d =0 sign (cid:16)Z Z χ Q r ( t ,...,t d ) (cid:17) r ∆ Q r ( t ,...,t d ) ( x ) , where Q r ( t , . . . , t d ) is the cube with side length r whose vertex closest to the originis ( rt , . . . , rt d ). Note that f is 2-Lipschitz.We can think of the number of points landing in each subcube as N independent λ = 1 Poisson processes which we stop once their sum reaches roughly P ( ζ i ∩ [0 , d = 0) | I | . By the law of large numbers, the stopping time will be very close to t = N − P ( ζ i = ± [ y ] , y ∈ [0 , d ) | I | and very nearly N te − t of the subcubes will contain exactly one point. Of these,with high probability, at least (1 / · − d · N te − t will be contained in the middle third of their subcube. Therefore, Z Z f ≥ c n,k L · N d − d with high probability . On the other hand, given j = 1, i ∈ I j , and i ′ ∈ I such that ζ i is correlated with ζ i ′ , Lemma 6.7 tells us that (cid:12)(cid:12) E (cid:0)R ζ i g i ′ (cid:1)(cid:12)(cid:12) ≤ C n,k r / and therefore (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) ≤ P i ∈ I j LC n,k r / ≤ LC n,k N d − d − d . Since this is small compared to R Z f , this shows that E (cid:16)Z Z ∩ P ~x f (cid:17) ≥ c n,k L N d − d Lip f for large enough N. Case d = 2 . This case is broadly similar, but we build the function f in a morecomplicated way, following the original proof of [AKT]. For an integer r , let Q rst bethe square of side length 2 − r whose lower left corner is at ( s · − r , t · − r ), and write∆ rst = ∆ Q rst . We write g ( x, y ) = . N X r =1 2 r − X s,t =0 g rst ( x, y ) = . N X r =1 2 r − X s,t =0 ∆ rst ( x, y ) Z Z ∆ rst = . N X r =1 X i ∈ I g rζ i ( x, y )= . N X r =1 X i ∈ I ∆ rv i ( x, y ) Z ζ i ∆ rv i where ζ i = ± [ v i ] . Notice that R Z g rst is always nonnegative: roughly speaking, it measures thesquare of the “imbalance” of positive and negative points in Q rst . In particular, it’snot hard to see that E ( R Z g rst ) = c n,k · − r N , and therefore E ( R Z g ) = c n,k N log N .On the other hand, the derivative of g is O ( √ N log N ) on average, but can bemuch larger in some places. To remedy this, Ajtai, Koml´os, and Tusn´ady introduceda “stopping time” rule, building f as the sum of some, but not all of the g rst . We donot need to give the exact definition, remarking only that the function f satisfies E (cid:0)R Z f (cid:1) ≥ c n,k L N log N (6.13) Lip f ≤ C n,k p N log N . (6.14)Now, for j = 1, (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) ≤ X i ∈ I j . N X r =1 X(cid:8)(cid:12)(cid:12) E (cid:0)R ζ i g rζ i ′ (cid:1)(cid:12)(cid:12) | ζ i ′ is correlated with ζ i (cid:9) . By Lemma 6.7, the value of each term of this triple sum is O (2 − r/ ). Therefore, (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) ≤ C n,k LN . N X r =1 − r/ ≤ (1 + √ C n,k LN.
ILLING RANDOM CYCLES 19
Combining this with (6.13) and (6.14), we see that E (cid:16)Z Z ∩ P ~x f (cid:17) ≥ c n,k L p N log N Lip f for large enough N. Case d = 1 . We split the interval [0 ,
1] into R equal regions, with R to be determinedlater. Write ∆ s = ∆ [ s/R, ( s +1) /R ] , and let g ( x ) = R − X s =0 ∆ s ( x ) R Z χ [ sR , s +1 R ] = X i ∈ I g ζ i ( x ) = X i ∈ I ± ∆ /Ry i ( x ) where ζ i = ± [ y i ] . We obtain the desired function f by replacing R Z χ [ sR , s +1 R ] with h s = sign (cid:18)Z Z χ [ sR , s +1 R ] (cid:19) min (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z Z χ [ sR , s +1 R ] (cid:12)(cid:12)(cid:12)(cid:12) , C n,k r NR (cid:27) for some sufficiently large C n,k . Then Lip f ≤ C n,k √ N R and R Z f ≥ c n,k L N .On the other hand, for j = 1 and any i ∈ I j and i ′ ∈ I such that ζ i and ζ i ′ arecorrelated, by Lemma 6.7, (cid:12)(cid:12) E (cid:0)R ζ i g ζ i ′ (cid:1)(cid:12)(cid:12) ≤ C n,k R − / , and therefore (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) ≤ C n,k LN R − / . For some large enough R , depending on n and k but not on N , (cid:12)(cid:12) E (cid:0)R Z j f (cid:1)(cid:12)(cid:12) ≤ L E (cid:0)R Z f (cid:1) . Thus E (cid:0)R Z ∩ P ~x f (cid:1) ≥ C n,k √ N Lip f , completing the proof. (cid:3) Concentration of measure
In this section, we show that when n − k ≥
2, the size of the filling tends toconcentrate around its mean. That is, we show that (1.2) holds in the case ofTheorems A, B, and C. We first prove this in the case of Theorem 3.1. The maintool is the concentration of measure in high-dimensional balls, an idea due to Gromovand Milman [GM] and of wide importance in probability theory [Led]. We follow theexposition due to Bobkov and Ledoux [BL19a, § Theorem 7.1.
Let Z be a random cycle in C ([0 , n , ∂ [0 , n ) as in Theorem 3.1.Then for every r > , P [ | F V ( Z ) − E ( F V ( Z )) | ≥ r ] ≤ C exp (cid:0) − C r/ √ N (cid:1) for universal constants C , C > . In particular, the standard deviation of
F V ( Z ) is at most O ( √ N ). In otherwords, for n ≥ F V ( Z ) / E ( F V ( Z )) converges to 1 as N → ∞ . Proof.
Equip X = C ([0 , n , ∂ [0 , n ) with the metric d F V ( Z, Z ′ ) = F V ( Z − Z ′ )and let E = [ − , × [0 , n − . Define ζ : E → X by ζ ( ± x , x , . . . , x n ) = ± [( x , x , . . . , x n )] and ζ : ( E N , d Eucl ) → X by ζ ( v , . . . , v N ) = N X i =1 ζ ( v i ) . This map is √ N -Lipschitz since when every point moves by a tiny amount ε , thedistance is √ N ε in the domain and
N ε in the range.Define the concentration function of a metric measure space (
M, d, µ ) of totalmeasure 1 to beconc M ( r ) = sup { − µ ( N r ( A )) | µ ( A ) ≥ / } , r > , where N r ( A ) is the r -neighborhood of the set A . The key observation of Gromovand Milman [GM, Thm. 4.1] is thatconc M ( r ) ≤ e − ln(3 / λ r , where λ is the first nonzero eigenvalue of the Laplacian on M . Since the spectrumof a product of manifolds is the sum of its spectra, λ is constant on powers of M .The map ζ is measure-preserving, so it follows thatconc X ( r ) ≤
34 exp (cid:0) − ln(3 / λ r/ √ N (cid:1) . Therefore, for any 1-Lipschitz function u : X → R , P [ | u ( Z ) − median( u ) | ≥ r ] ≤
32 exp (cid:0) − ln(3 / λ r/ √ N (cid:1) . By Chebyshev’s inequality, the same, modulo constants, holds for the mean (seealso [Led, Prop. 1.10]). (cid:3)
To adapt this proof for the case of Theorems A, B, and C, we just have to changethe space E : take E sphere = f Gr k ( R n ) for Theorem A E cube = { affine k -planes P ⊂ R n | P ∩ [0 , n = ∅} for Theorem B E knot = [0 , n for Theorem C . In the first two cases, the map ζ is constructed as before. In the last case, for a k -pseudomanifold M with vertex set M , ζ M : E | M | knot → Z k ([0 , n ) sends ( v , . . . , v | M | )to the image of the linear immersion of M with vertices ( v , . . . , v | M | ).In each case, it is easy to see that if the space of k -cycles is given the filling volumemetric, then ζ is √ N -Lipschitz. Therefore, the rest of the proof is identical to thatof Theorem 7.1. References [ABD + ] J. Arsuaga, T. Blackstone, Y. Diao, E. Karadayi, and M. Saito, Linking of uniform randompolygons in confined spaces , J. Phys. A (2007), no. 9, 1925–1936.[AKT] Mikl´os Ajtai, J´anos Koml´os, and G´abor Tusn´ady, On optimal matchings , Combinatorica (1984), no. 4, 259–264.[Bany] Augustin Banyaga, Formes-volume sur les vari´et´es `a bord , Enseign. Math. (2) (1974),127–131. ILLING RANDOM CYCLES 21 [BL19a] Sergey Bobkov and Michel Ledoux,
One-dimensional empirical measures, order statis-tics, and Kantorovich transport distances , Mem. Amer. Math. Soc. (2019), no. 1259,v+126.[BL19b] ,
A simple Fourier analytic proof of the AKT optimal matching theorem , arXivpreprint arXiv:1909.06193 (2019).[BMPR] Martins Bruveris, Peter W. Michor, Adam Parusi´nski, and Armin Rainer,
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