aa r X i v : . [ m a t h . AG ] J u l Noncommutative geometry of random surfaces
Andrei Okounkov
This paper is about a certain interaction between probability and geometry.The random objects involved will be random stepped surfaces spanning agiven boundary in R . Equivalently, one can talk about random rhombitiling of a planar domain or random dimer coverings of certain subgraphs ofthe hexagonal graph. Probabilistic questions about these random surfaceswill be answered in terms of a nonrandom algebraic object, geometrically acurve in a noncommutative plane.Underlying this connection is Kasteleyn’s theory of planar dimers whichcomputes all probabilities in terms of the Green’s function of a certain finite-difference operator K . It will be clear from our construction that the connec-tion between finite-difference operators and noncommutative geometry maybe easily extended far beyond what we do in this paper. Our goal here, how-ever, is to explain a certain phenomenon in the least possible generality andto stay as close as possible to certain specific applications. These applica-tions, as well as some other directions that look promising will be discussedbelow. Let Ω be a simply-connected planar domain that can be tiled by rhombi asshown in Figure 1. Well-known bijections illustrated in Figure 1 identify suchtilings with dimer coverings of subgraph Ω = Ω ∩ Γ of the hexagonal graphΓ and also with stepped surfaces spanning given boundary. An introductionto dimers and stepped surfaces may be found in [8].1igure 1: Stepped surfaces are the same as tilings of a planar domain Ω byrhombi. Each tile is a union of a black and white triangle. The adjacency graphof the triangles is (a piece of) the 6-gonal graph.
Stepped surfaces arise in mathematical physics in a variety of contexts:from simply-minded, but realistic models of interfaces (e.g. crystalline sur-faces) to the sophisticated setting of super-symmetric gauge and string the-ories (see e.g. [14, 16] for an introduction).In all applications, it is natural to weight the probability of a steppedsurface S by the volume V ( S ) enclosed by it, i.e. to setProb( S ) ∝ q V ( S ) , (1)where q > S and S spanningthe same boundary the difference V ( S ) − V ( S ) is well-defined, which is allthat matters in (1). In the crystal surface context, log q is the energy pricefor removing an atom We will be particularly interested in the case when Ω grows to infinity, whilekeeping its shape, that is, the number and orientation of its boundary seg-ments. We will refer to such domains as polygonal . The study of a randomstepped surface a large polygonal boundary (equivalently, random dimer cov-ering, or random tiling) leads to interesting probabilistic questions. Thenontriviality of these questions may be appreciated by looking at Figure 2. Note that there is a well-known ambiguity in reconstructing 3-dimensional surfaces
A simulation of the limit shape formation. The curve separating theordered regions (facets) from the disorder is called the frozen boundary . Here, itis a cardioid, i.e. the dual of a rational cubic.
Apparent in Figure 2 is a formation of a certain nonrandom limit shape ,in other words, as the mesh size goes to zero so does the scale of randomness.This is a form of the law of large numbers. The existence of a limit shape wasproven for stepped surfaces (with arbitrary boundary conditions) by H. Cohn,R. Kenyon, and J. Propp in [4].In [9], the limit shape for polygonal boundaries was linked to a certainplane algebraic curve Q . In particular, the frozen boundary, which is thecurve separating order from disorder in Figure 2, is the planar dual of Q inexponential coordinates. The main object of this paper may be characterized, informally, as a quanti-zation of the limit shape, or of the curve Q to be more specific. This quanti-zation exists for finite Ω, that is, before any limits are taken. In particular,it captures not just the limit shape but also the fluctuations of our random from tiling, namely, one can switch the roles of convex and concave corners. A rotationby π/ q and not to the size of fluctuations (as couldbe expected from the uncertainty principle). In principle, classical Kasteleyn’s theory [6] answers all possible questionsabout random stepped surfaces in terms of the Green’s function, i.e. the in-verse of a certain difference operator. This
Kasteleyn operator K is a weightedadjacency matrix of the graph Ω .Note that Γ , and hence, Ω is bipartite, that is, its vertices may becolored in two colors (black and white, traditionally) so that only verticesof opposite color are joined by an edge. This is reflected in Figure 3. Wewill index the rows (resp. columns) of the adjacency matrix by white (resp.black) vertices. The nonzero matrix elements K ij should satisfy q K K K = K K K (2)for each face F , where v , . . . , v are the six vertices going around F as inFigure 3. This fixes K uniquely up to a certain gauge transformation, namelyleft and right multiplication by a diagonal matrix.
12 3 4 56
Figure 3:
Each hexagon face carries log q units of magnetic flux. The goal of this paper may be informally described as looking for somehidden structures in the inverse matrix K − . Certain structures in K − areplain to see: by definition, the entries of K − satisfy a finite-difference equa-tion in each index, namely K K − = K − K = 1.4ur main claim is that for polygonal domains Ω the entries of K − satisfy additional finite-difference equations. The degree of these additional equa-tions is determined by the shape of Ω, that is, by the number of boundarysegments, and not by the size of Ω. This is crucial from the probabilisticviewpoint. The noncommutative geometry of the title provides a natural language tostate and study these additional equations.The origin of the noncommutativity may be traced to (2). For q =1, the adjacency matrix of Γ is an obvious solution and this solution istranslation-invariant, i.e. commutes the the subgroup Z ⊂ Aut(Γ ) actingby (bipartition-preserving) translations.For q = 1, the translation-invariant equation (2) has no translation-invariant solutions, which means that the Kasteleyn operator K now com-mutes with magnetic translations , i.e. translations followed by a gauge trans-formation. In turn, magnetic translations commute only up to a factor(whose logarithm is proportional to the area of the parallelogram spannedby the translation vectors). They form, in other words, an algebra known asthe quantum 2-torus.The considerations so far apply only to the whole 6-gonal graph Γ , thatis, in the absence of any boundaries. Somewhat remarkably, however, a cer-tain framework may be established in which the Kasteleyn operator and thecommuting magnetic translations act in a way compatible with polygonalboundaries. This involves compactifying the quantum 2-torus to a noncom-mutative plane, meaning that one introduces a certain graded algebra A , adeformation of the ring of polynomials in x , x , x , such that the quantum2-torus is the degree 0 part of A [( x x x ) − ]. For any fixed white vertex w ∈ Ω , the action of magnetic translation on K − ( · , w ), that is, on the corresponding column of K − , yields a graded A -module Q w . The additional equations satisfied by K − will be reflected in thefact that Q w is a torsion module.For different w , the modules Q w share the same fundamental features. Infact, there is a canonical submodule Q in all of them that depends on Ω only5nd captures the essential information. The construction of the module Q and the study of its basic properties willoccupy the bulk of the present paper. While the definition of Q involvesnothing beyond elementary combinatorics and linear algebra, we will findthe resulting object has a certain depth and complexity.The degrees of its generators and relations (and hence the degrees of theadditional equations satisfied by K − ) are determined by the combinatoricsof the domain Ω only, see Theorem 1. On the other hand, the explicit formof these relations depends of q and the geometry of Ω in a rather intricatefashion. This is where a geometric way of thinking about such modules, pioneeredby M. Artin and his collaborators, becomes essential (see e.g. [21] for anintroduction). While its a matter of definitions to associate to Q a sheafon the noncommutative plane, the geometric intuition thus gained is veryvaluable.In the first place, this is what allows us to view Q as a quantization of thelimit shape Q . In fact, quantization requires additional degrees of freedom,parametrized by line bundles (and more general rank 1 sheaves) on Q . Thesemay be compared to complex phases in quantum mechanics.Certain features of Q , such as its points of intersection with the coordinateaxes of P have a direct quantum analog satisfied by Q , see Section 4. Others(such as the geometric genus) are harder to generalize, see [10]. Because Ω is a purely combinatorial object, we can modify it by simplymoving boundary segments in and out. We prove in Section 5 that thesetransformations act on Q by what may be called a noncommutative shift onthe Jacobian. In particular, this describes what happens to Q under rescalingof Ω. 6 .11 Acknowledgments The author’s present understanding of the subject took some time to developand I have a large number of people to thank for stimulating and insightfuldiscussions along the way. In particular, I thank D. Eisenbud, V. Ginzburg,A. J. de Jong, R. Kenyon, I. Krichever, D. Maulik, N. Nekrasov, E. Rains,N. Reshetikhin, and Y. Soibelman. Parts of this work grew into joint researchprojects [10, 19].I had an opportunity to lecture on the subject on a number of occasions,in particular during the Aisenstadt lectures at the Universit´e de Monr´eal,T. Wolff lectures at Caltech, Eilenberg lectures at the Columbia University,and Milliman lectures at the University of Washington. I am very gratefulto the participants of these lectures for their involvement and feedback. Ivery much thank these institutions and the Institut des Hautes ´Etudes Sci-entifiques for their warm hospitality during my work on this paper.
Let A be the algebra generated by x , x , x subject to the relations x j x i = q ij x i x j , (3)where q ii = 1 and q ij = q − ji . This is a basic example of a noncommutativeprojective plane P , see e.g. [21] for an introduction. Adding the inverses x − i of the generators, we obtain a larger algebra T known as a noncommutative3-torus.Both A and T are graded by the total degree in all three generators.Let T d , where d ∈ Z is arbitrary, be one of the graded components. Themonomials x a = x a x a x a ∈ T d , obviously correspond to lattice points a ∈ Z lying in the plane a + a + a = d . Their projection in the (1 , ,
1) direction, that is, their image in R / R · (1 , , ∼ = R may be identified with the black vertices of Γ . In thesame fashion, the monomials in T d +1 are put into bijection with the whitevertices of Γ . This is illustrated for d = 4 in Figure 4.7igure 4: Black and white vertices may be identified with points of Z lying ontwo parallel hyperplanes and thus with monomials of two successive degrees. Herewe only plot points in R ≥ which correspond to polynomials in the x i ’s. Consider the operator K : T d → T d +1 given by right multiplication by x + x + x ∈ W , that is, K f = f · ( x + x + x ) . One easily checks from the commutation relations (3) that, indeed, in thebasis of monomials, this is a q -weighted Kasteleyn operator for Γ as above,with q = q q q . (4)Note that while there is no canonical ordering of the variables and, hence,no canonical normalization of a monomial, the lines spanned by monomialsin T are well defined. This is all we need since we only care about K modulogauge transformations. M Monomials are normal in T , that is, A x c = x c A is naturally a A -bimodule.Assuming deg x c ≤
1, monomials in ( A x c ) form a triangle with vertices x c x c x − c − c +13 , x c x − c − c +12 x c , x − c − c +11 x c x c ∈ T .
8e denote this equilateral triangle by ∆( c ) and call it the support of ( A x c ) . We think of black and white vertices in Ω as monomials in T and T ,respectively. Write Ω as a set-theoretic combination of trianglesΩ = [ i ∆( a i ) \ [ j ∆( b j ) , a i , b j ∈ Z , (5)as in Figure 5. The inclusion of triangles induces the inclusion of graded A -bimodules M A x b i ⊂ M A x a i . (6)We denote by M the quotient A -bimodule in (6). Define an endomorphism K of M as the right multiplication by x + x + x ∈ A . This is a map ofleft A -modules. By construction, the operator K : M → M is the Kasteleyn operator for Ω .Figure 5: The domain Ω may be represented as a difference of unions of triangles.Here, it is the union of 2 blue triangles minus the union of 4 green ones. .3.3 The idea of stable range will play an important part in this paper. Bydefinition, a number i is in the stable range for M if the domainΩ( i ) = supp M i +1 has the same combinatorics as Ω = Ω(0). In particular, the length of theshortest white boundary of Ω, i.e. a boundary formed by white triangles,gives an upper bound on the stable range.More generally, refining (5), M may be given a resolution0 → F → F → F → F → M → L A x a i , where the maps are the natural inclusions.Here F = L A x a i corresponds to the triangles ∆( a i ) in (5). The intersec-tions among A x a i together with ∆( b i ) contribute to relations F , and so on.The module F doesn’t have generators of positive degree, but the other F i ’sdo. The stable range is until the first such generator appears.Note that the stable range scales linearly with Ω, meaning that it scaleslike the inverse mesh size in the probabilistic setup. Throughout the paper,we will only be interested in what happens in the stable range. Our next goal is to compute the Hilbert function of M d in the stable range.For that, we will make a genericity assumption that all 3 boundary slopesappear in the boundary ∂ Ω of Ω in a cyclic order. We will denote by deg Ω thenumber of times ∂ Ω cycles through the three slopes. For example, deg Ω = 3in Figure 6.
Lemma 1.
For d ≥ in the stable range, we have dim M d = dim M − deg Ω d ( d − − d ind K . Here, of course, the index of K equals zero, but it will not vanish in thegeneralizations considered below. Proof.
Consider the map M d ∋ f x f ∈ M d +1 . (8)10he kernel and cokernel of this map are formed by functions supported onhorizontal strips of the form shown in Figure 6. More precisely, white hori-zontal boundaries correspond to the cokernel, the black ones — to the kernel.Figure 6: A white (left) and black (right) boundary strips. Note that they alwayshave the same shape provided the boundary of Ω cycles through the 3 directions.
Note that white boundaries of supp M i shrink with i , while the black onesexpand. Thus each of the deg Ω horizontal boundaries contributes − M i with respect to i . We concludedim M d = − deg Ω d . . . , . Here dots stand for a polynomial in d of degree ≤
1, which is uniquely fixedby its evaluation at d = 0 , The operator K : M d → M d +1 is surjective for d ≥ in the stablerange and q > or q generic.Proof. For d = 0, by Kasteleyn’s theorem, det K determinant gives the q -weighted count of stepped surfaces, hence nonzero for q >
0. For d > K on a white boundary strip, as in Figure 6, whichis immediate. Q and the inverse Kasteleyn matrix We denote by Q the kernel of K acting on M . This is a graded left A -module.To see why this may be a useful definition, let us generalize the constructionslightly. 11 .4.2 Let w be a white vertex of Ω corresponding to a monomial x w . Let Ω w beobtained from Ω by removing the corresponding white triangle and set M w = M (cid:14) A x w . The conclusions of Lemmas 1 and 2 continue to hold for M w , with the obviousmodification that deg Ω w = deg Ω + 1 , ind K w = 1 . In contrast to Q , we have Q w = 0. Indeed, by construction, Q w is spannedby the corresponding column K − x w of the inverse Kasteleyn matrix.Via the left A -module structure on Q w , the algebra A acts on the columnsof the inverse Kasteleyn matrix by difference operators. To see that a nonzerodifference operator from A must annihilate Q w , it will suffice to compute theHilbert function of Q w . From Lemmas 1 and 2 we have, in the stable range,dim Q d = d deg Ω + ind K = d deg Ω (9)and, similarly, dim Q w d = (deg Ω + 1) d + 1 . (10)Since this dimensions grow only linearly in d , for any g ∈ Q d the map A i ∋ f f · g ∈ Q d + i must have a kernel as soon as i is large enough. These are the sought differ-ence equations satisfied by K − . We will, obviously, have do more work tosay something more specific about them. Note that we have an exact sequence0 → Q → Q w → L → , (11)12here the third term L satisfies dim L d = d + 1. In fact, L is a line module ,i.e. the module of the form L = A (cid:14) A l , l ∈ A . Lemma 3.
In the stable range, i.e. for w sufficiently far from the boundaryof Ω , L is a line module with l = x w ( x + x + x ) x − w .Proof. Let g = K − x w be the generator of Q w and suppose f g ∈ Q for some f ∈ A . This means that f g may be extended to all of Ω as a solution ofKasteleyn’s equation. In other words, there exists a polynomial g ′ ∈ A deg f − such that 0 = K ( f g + g ′ x w ) = f x w + g ′ x w ( x + x + x ) , proving the assertion.From this perspective, there isn’t much difference between Q and Q w .Somewhat poetically, we will call Q the quantum limit shape . Mathemat-ical reasons for this name will be discussed below. Q The goal of this Section is to prove the following
Theorem 1.
In the stable range and for generic q , Q is generated by deg Ω generators of degree subject to deg Ω linear relations. In other words, theminimal graded free resolution of Q has the form → A ( − deg Ω → A ( − deg Ω → Q → . (12) Similarly, → A ( − deg Ω → A ⊕ A ( − deg Ω − → Q w → . (13)Here we denote, as customary, A ( i ) d = A i + d .13 .2 In the commutative case, resolutions of the form (12) are well known inalgebraic geometry, see, in particular, [1]. The corresponding sheaves on P are of the form ι ∗ L , where ι : Q → P is an inclusion of a curve of degree D = deg Ω and L is a line bundle L (ora more general torsion-free sheaf in case Q is singular) of degree g −
1. Here g = ( D − D − / Q . Concretely, Q = det R , where R is the matrix of linear forms that gives the map R : A ( − D → A ( − D in (12). The condition in (12) on L to have no sections means L ∈ Jac g − ( Q ) \ Θ, where Θ ⊂ Jac g − ( Q ) is the theta divisor.The meaning of (13) is parallel, with the difference that D becomesdeg Ω + 1 and deg L becomes g . In the commutative case, (11) implies thesupport of Q w has the line l = 0 as a component. One of the eventual goals of the present project is to understand the behaviorof Q as the mesh size goes to 0 while log q → P .It will be shown in [10] that this limit is supported on the curve Q cor-responding to the limit shape. This is the reason for calling Q the quantumlimit shape. For any graded A -module Q , the degrees of its generator, relations, etc., maybe read off the dimensions of the graded components of the vector spacesTor i Q = Tor i ( A / A > , Q ) , A > is the ideal generated by A . In particular, the existence of a freeresolution of length 2 is equivalent to the following Lemma 4.
In the stable range of degrees,
Tor i Q = 0 for i > .Proof. Since Tor > vanish identically for the algebra A , we need to show thevanishing for i = 2 ,
3. Since the resolution (7) is defined combinatorially interms of Ω, we conclude Tor i M = 0 in the stable range. This is, really, thedefinition of the stable range. Since Q ⊂ M and Tor = 0 identically, weconclude Tor Q = 0 . By construction, there is an exact sequence0 → Q → M → Im K → , and since Im K ⊂ M , we have Tor Im K = 0. Now from the long exactsequence for Tor i , it follows that Tor Q = 0. Lemma 5.
For generic q , Q is generated by Q .Proof. It suffices to consider the commutative case q ij = 1. Then, on the onehand, Q is annihilated by x + x + x , while on the other Tor i Q = 0, i > Q corresponds to a vector bundle on the line x + x + x = 0.From its Hilbert polynomial, we see that it must be O ( − deg Ω , whence theconclusion. Now it is easy to complete the proof of (12). We have deg Ω = dim Q generators in degree 1 and from (9) we see that they must satisfy deg Ωlinear relations. There are no other generators by Lemma 5 and no otherrelations by (9).The proof of (13) goes along the same lines. Lemma 4 still holds eventhough (Tor M w ) = 0. The analog of Lemma 5 is that Q w is generatedin degrees 0 and 1, because in the commutative case Q corresponds to thebundle O ⊕ O ( − deg Ω on the line x + x + x = 0.15t remains to explain why the commutative resolution0 → A ( − ⊕ A ( − deg Ω → A ⊕ A ( − deg Ω → Q w → , q ij = 1 , jumps to (13) for generic q . In other words, we need to check that thegenerator of Q w no longer satisfies a linear relation for generic q . This is aneasy consequence of the results of the next section.Namely, a linear polynomial meets x x x in 3 points, that is, it annihi-lates 3 generators of point modules over A . As we will see, the annihilator of Q w meets each coordinate axis in deg Ω + 1 points. Recall from [9] that the curve Q describing the limit shape is determinedas the unique rational curve of degree deg Ω for which the dual curve Q ∨ isinscribed in Ω. This means, in particular, that Q meets each coordinate lineof P in deg Ω specified points.In this section, we will see that the quantum limit shape Q satisfies theexact noncommutative analog of this incidence. We define ∂ Q = Q (cid:14) x Q . We will see that ∂ Q is a direct sum of deg Ω point modules labeled by thehorizontal boundaries of Ω as in Figure 6.By definition, the Hilbert polynomial of a point module is equal to theconstant 1. Up to modules of finite length, point modules are parametrizedby the toric divisor x x x = 0 of P . The correspondence is a follows( a : a : 0) ↔ A (cid:14) h x , a x − a x i . The ratios a /a for the summands of ∂ Q will be determined by the verticalcoordinate of the horizontal boundaries of Ω.16 .3 The Hilbert function evaluationdim ( ∂ Q ) d = deg Ω , d > , is a consequence of the following Lemma 6.
Left multiplication by x has no kernel acting on Q .Proof. A polynomial in the kernel of left multiplication by x has a supportin a strip of width 1 along the black boundaries, as in Figure 6, right. It isimpossible for such function to be annihilated by the Kasteleyn operator. Let Ω ′ be a nontileable domain obtained by moving one of the white horizon-tal boundaries of Ω one step in. Denote by M ′ the corresponding monomialmodule. Let K ′ : M ′ → M ′ be right multiplication by x + x + x and let Q ′ = Q ∩ M ′ be its kernel. Since x Q ⊂ M ′ , ∂ Q surjects onto Q / Q ′ .By cutting white boundary strips off Ω ′ ( i ), i >
1, we see as in the proofof Lemma 2 that K ′ surjects onto M ′ i for i >
1. Since ind K ′ = −
1, it followsfrom Lemma 1 that Q / Q ′ is a point module. Lemma 7.
Let a be the vertical coordinate of the strip Ω \ Ω ′ , that is, let deg x f = a for all f ∈ M/M ′ . Then Q / Q ′ ∼ = A / h x , x + q a x i (1) , starting in degree .Proof. The series( x + x ) − = x − − x − x x − + x − x x − x x − − . . . gives 1 after left or right multiplication by ( x + x ). Interchanging the rolesof x and x and taking the difference δ = ( x + x ) − − ( x + x ) − we get an analog of the usual δ -function, which is annihilated by both leftand right multiplication by ( x + x ).17n element of ( Q / Q ′ ) d is a truncation of the series x a x − a + d +11 δ on bothsides. Left multiplication by x + q a q d +121 x annihilates it, whence the conclusion. Now let Ω ′ be a nontileable domain obtained by moving one of the blackhorizontal boundaries of Ω one step out. We have a map M ′ → M corre-sponding to the restriction of functions and from Lemma 6 we conclude thatit yields an injection Q ′ → Q . Further, x Q is in the image of Q ′ , hence Q / Q ′ is again a point module onto which ∂ Q surjects. Lemma 8.
Let a be the vertical coordinate of the strip Ω ′ \ Ω , that is, let deg x f = a for all f in the kernel of the restriction map M ′ → M . Then Q / Q ′ ∼ = A / h x , x + q a x i (1) , starting in degree .Proof. Let f be in Q d and denote by f d − a +1 x a − the monomials in f alongthe boundary in question. Here f d − a +1 denotes a polynomial in x and x ofdegree d − a + 1. Clearly, f may be extended to an element in Q ′ if and onlyif we can find a polynomial g d − a ( x , x ) x a with support in Ω ′ \ Ω such that f d − a +1 x a = g d − a x a ( x + x ) . The commutation relations in A imply that for any f k ∈ A k which does notdepend on x and any C we can find f ′ k ∈ A k such that( x + C x ) f k = f ′ k ( x + q k C x ) . From this it follows that x + q a q d +121 x annihilates ( Q / Q ′ ) d , as was to beshown. We can summarize the discussion as follows.18 heorem 2.
We have ∂ Q ∼ = deg Ω M i =1 A / h x , x + q a i x i (1) , (14) starting in degree , where a i are the heights of the horizontal boundaries of Ω as above.Proof. The Hilbert polynomial equals the constant deg Ω on both sides. Weconstructed a map from the LHS to the RHS in (14). Its kernel consists offunctions f that vanish on the boundary strips along the white boundariesof Ω and may be extended as solutions of K f = 0 to a horizontal strip justbeyond the black boundaries of Ω. This modified domain is a translate ofΩ( −
1) in the x direction and our conditions on f imply f ∈ x Q . Thus themap above is an isomorphism. Let Ω ′ be obtained from Ω by moving one boundary segment by one step, asin Sections 4.4 and 4.5. We found that the corresponding modules Q and Q ′ fit into an exact sequence of the form0 → Q ′ → Q → P → P . Modules of the form (12) form an open set in themoduli spaces of A -modules of rank 0 and( c , c ) = ( D, D ( D + 3) / , D = deg Ω . Let M ( c , c ) denote a finite cover of this open set over which the orderingof the summands in (14) is chosen.For general c , we define M ( c , c ) as the moduli space of A -modules withthe same minimal resolution as the generic commutative resolution plus anordering of boundary points. For example, for( c , c ) = ( D, D ( D + 3) / − k ) , ≤ k ≤ D/ , the corresponding modules Q ′′ are of the form0 → A ( − D − k → A k ⊕ A ( − D − k → Q ′′ → , generalizing (13). 19 .2 Noncommutative shift on the Jacobian It is easy to see that dim M ( c , c ) = c + 1 , and, in particular, it doesn’t depend on c . Lemma 9.
The correspondence { ( Q ′ , Q ) } ⊂ M ( c , c + 1) × M ( c , c ) defined by (15) is the graph of a birational map.Proof. It suffices to consider the commutative case, when this becomes a shiftby P on the Jacobian of the curve Q , see Section 3.2.This means that we have an action of of a group S ∼ = Z c on F c M ( c , c )by birational transformations. A subgroup of the form Z c − preserves c and acts birationally on individual components. Perhaps the title of thissubsection is the appropriate name for this group action.Parallel group actions may be defined for other noncommutative surfaces.They turn out to encompass several previously studied discrete dynamicalsystems. This is the subject of a joint work by E. Rains and the author, theresults of which will appear in [19]. Renormalization is a central concept in mathematical physics. For anytileable domain Ω and any m ∈ Z > , the scaled domain m Ω is again tileableand it is natural to ask how this scaling transformation affects our randomsurfaces. Equivalently, of course, one can keep Ω fixed and divide the meshsize by m .To quantify the word “affects”, one tries to summarize the behavior ofrandom surfaces in terms of finitely many essential degrees of freedom. Onefurther hopes to define an action of the group R > on this space extendingthe scaling transformations above.While this procedure is a very powerful guiding principle, in practice oneusually has to use various approximations to make it work. This is onlynatural since one is trying to squeeze an infinite-dimensional problem into afinite-dimensional dynamical system.20 .4 Our kind of problems are special in that they have a certain built-in finite-dimensionality, starting from a finite number segments that bound Ω. Thequantum limit shape Q also varies in a finite-dimensional moduli space. Since Q encodes the essential information about the correlation functions, therenormalization dynamics may be considered understood once the scalingaction on Q is determined.Clearly, scaling transformations are composed out of many noncommuta-tive shifts on the Jacobian, and more precisely we have Theorem 3.
The scaling Ω m · Ω preserves the S -orbit of Q and inter-twines the action of s ∈ S with the action of m · s ∈ S . If the parameter q is adjusted simultaneously, then the limit m → ∞ becomesthe thermodynamic limitmesh → , log q = O (mesh) , which is the limit that we were planning to take all along. In this limit,noncommutative shifts of the Jacobian may be viewed as a perturbation ofthe commutative shift, thus joining a much-studied area of perturbations ofintegrable systems. Their analysis from this point of view will appear in [10].In particular, it will be shown in [10] that, indeed the quantum limitshape Q is a deformation of the curve Q that defines the (classical) limitshape. The Kasteleyn operator on Γ has an infinite-dimensional kernel and theKasteleyn equation needs to be supplemented by boundary conditions inorder to have a unique solution. By contrast, once a second difference equa-tion of degree d is known, the solutions form a d -dimensional linear space,spanned by modulated plane waves. This simple principle gives a powerful21ay to control K − in the thermodynamic limit which is the basic analyticissue in the analysis of stepped surfaces.In fact, optimistically, one may expect these techniques to overcome thedifficulties that lie in the way of proving the CLT for stepped surfaces withpolygonal boundaries (see [7] for techniques that can handle a different sortof boundary conditions) as well determining the local correlations. Further,since polygonal boundaries are dense in the space of all boundaries, the calcu-lation of local correlations for them has direct implication to the classificationof Gibbs measures (about which [20] contains a wealth of information). Kasteleyn theory and the formalism of [9] work for any periodic bipartiteplanar dimer. It would be very interesting to study the quantum limit shapein this generality. It would be also very interesting to find applications todifference equations other than the Kasteleyn equation, such as e.g. discreteDirac equations in dimensions > For very special kinds of boundary conditions (see e.g. [16, 17] for an in-troduction) the q -weighted stepped surface partitions functions Z becomegenerating functions Z DT for the Donaldson-Thomas invariants of toric CYthree-folds .Very generally, for any smooth projective three-fold X the expansion oflog Z DT in powers of log q was conjectured to generate the Gromov-Witteninvariants of the same three-fold X , genus by genus [11]. For a toric three-fold X , this conjecture was proven in [12].In particular, this relates the genus 0 Gromov-Witten invariants of X to the leading asymptotics of log Z as log q →
0, and hence to the limitshape Q . Mirror symmetry , which is certainly too complex and multifaceteda phenomenon to be discussed here with any precision, associates basicallythe same curve Q to X . Thus the limit shape point of view puts mirrorsymmetry on a firm probabilistic ground in this particular instance.The quantum limit shape Q captures the fluctuations and hence all ordersof the expansion of log Z . This makes it a strong candidate for the as yet The customary parameter q in DT theory differs from ours by a minus sign. X . Moreover, the quantum limit shape Q , being a categorical object, might stand a better chance of generalizationto non-toric X than the underlying box-counting. Progress in this directionremains both very desirable and scarce. There exists a different (and, at present, conjectural) way to extract thehigher genus GW invariants out of Q , which was proposed in [2] based onthe diagrammatic techiques developed in the random matrix context, see inparticular [3]. The use of noncommuting variables in a related context wasadvocated, in particular, in [5].It is reasonable to expect the two approaches to converge, especially sincevarious random matrix models may naturally be viewed as continuous limitsof stepped surfaces. For example, one sees random matrices quite directlynear the points where Q intersects the coordinate axes [18]. There are alsonumerous parallels between random matrices and Plancherel-like measureson partitions, which are induced on slices of stepped surfaces [15].A more ambitious goal may be to push the theory away from the K X = 0case. As a first problem in the K X = 0 direction, one can try the equivariantvertex [11, 12]. As a random surface model, it is very nonlocal and otherwisedistant from what is perceived as natural in statistical mechanics. To itscredit, it has a map, due to Nekrasov [13], onto certain local 2-dimensinallattice fermions. In contrast to Kasteleyn theory, these fermions are nowinteracting and it remains to be seen how much progress one can make inthis more general setting. References [1] A. Beauville,
Determinantal hypersurfaces , Mich. Math. J. , 39–64(2000).[2] V. Bouchard, A. Klemm, M. Marino, S. Pasquetti, Remodeling the B-model , Comm. Math. Phys. 287 (2009), no. 1, 117–178.[3] L. Chekhov, B. Eynard, N. Orantin,
Free energy topological expansionfor the 2-matrix model . J. High Energy Phys. 2006, no. 12.234] Cohn, H., Kenyon, R., Propp, J., A variational principle for dominotilings,
Journal of AMS , (2001), no. 2, 297-346.[5] R. Dijkgraaf, L. Hollands, P. Sulkowski, C. Vafa, Supersymmetric gaugetheories, intersecting branes and free fermions , J. High Energy Phys.2008, no. 2.[6] P. Kasteleyn,
Graph theory and crystal physics , Graph Theory and The-oretical Physics, 43–110, Academic Press, 1967[7] R. Kenyon,
Height fluctuations in honeycomb dimers , math-ph/0405052 .[8] R. Kenyon, Lectures on dimers , available from ∼ rkenyon/papers/dimerlecturenotes.pdf [9] R. Kenyon and A. Okounkov, Limit shapes and complex Burgers equa-tion , math-ph/0507007 .[10] I. Krichever and A. Okounkov, in preparation.[11] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory , I. & II., math.AG/0312059 , math.AG/0406092 .[12] D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande, Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds , arXiv:0809.3976 .[13] N. Nekrasov, Topological strings and two dimensional electrons , TheQuantom Structure of Space and Time, Proceedings of the 23rd SolvayConference on Physics, edited by D. Gross, M. Henneaux, A. Sevrin,World Scientific, 2007.[14] N. Nekrasov,
Instanton partition functions and M-theory , Vth TakagiLectures, Japan. J. Math. 4, 63-93 (2009)[15] A. Okounkov,
The uses of random partitions , XIVth InternationalCongress on Mathematical Physics, 379–403, World Sci., 2005.2416] A. Okounkov,
Random surfaces enumerating algebraic curves , Proceed-ings of Fourth European Congress of Mathematics, EMS, 751–768, math-ph/0412008 .[17] A. Okounkov,
Geometry and physics of localization sums , .[18] A. Okounkov, The birth of a random matrix , Mosc. Math. J. (2006),no. 3, 553–566.[19] E. Rains and A. Okounkov, in preparation.[20] S. Sheffield, Random surfaces , Ast´erisque (2005).[21] J. T. Stafford and M. Van den Bergh,
Noncommutative curves and non-commutative surfaces , Bull. AMS38