aa r X i v : . [ m a t h . P R ] A p r The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2012
ARCTIC CIRCLES, DOMINO TILINGS ANDSQUARE YOUNG TABLEAUX
By Dan Romik
University of California, Davis
The arctic circle theorem of Jockusch, Propp, and Shor assertsthat uniformly random domino tilings of an Aztec diamond of highorder are frozen with asymptotically high probability outside the“arctic circle” inscribed within the diamond. A similar arctic circlephenomenon has been observed in the limiting behavior of randomsquare Young tableaux. In this paper, we show that random dominotilings of the Aztec diamond are asymptotically related to randomsquare Young tableaux in a more refined sense that looks also at thebehavior inside the arctic circle. This is done by giving a new deriva-tion of the limiting shape of the height function of a random dominotiling of the Aztec diamond that uses the large-deviation techniquesdeveloped for the square Young tableaux problem in a previous pa-per by Pittel and the author. The solution of the variational problemthat arises for domino tilings is almost identical to the solution for thecase of square Young tableaux by Pittel and the author. The analytictechniques used to solve the variational problem provide a system-atic, guess-free approach for solving problems of this type which haveappeared in a number of related combinatorial probability models.
1. Introduction.
Domino tilings and the arctic circle theorem.
A domino in R isa Z -translate of either of the two sets [0 , × [0 ,
2] or [0 , × [0 , S ⊂ R is a region comprised of a union of Z -translates of [0 , , a domino tiling of S is a representation of S as a union of dominoes with pairwise disjointinteriors. Domino tilings, or equivalently the dimer model on a square lattice,are an extensively studied and well-understood lattice model in statisticalphysics and combinatorics. Their rigorous analysis dates back to Kaste-leyn [20] and Temperley and Fisher [30], who independently derived the for- Received October 2009; revised October 2010.
AMS 2000 subject classifications.
Key words and phrases.
Domino tiling, Young tableau, alternating sign matrix, Aztecdiamond, arctic circle, large deviations, variational problem, combinatorial probability,Hilbert transform.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2012, Vol. 40, No. 2, 611–647. This reprint differs from the original in paginationand typographic detail. 1
D. ROMIK (a) (b)
Fig. 1.
The Aztec diamond of order and one of its tilings by dominoes. mula m Y j =1 n Y k =1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) πjm + 1 (cid:19) + 2 √− (cid:18) πkn + 1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) / for the number of domino tilings of an n × m rectangular region. Aboutthirty years later, a different family of regions was found to have a muchsimpler formula for the number of its domino tilings: if we define the Aztecdiamond of order n to be the setAD n = n − [ i = − n min( n + i,n − i − [ j =max( − n − i − , − n + i ) [ i, i + 1] × [ j, j + 1](see Figure 1), then Elkies et al. [6] proved that AD n has exactly2( n +12 )domino tilings. This can be proved by induction in several ways, but isperhaps best understood via a connection to alternating sign matrices .One of the best-known results on domino tilings is the arctic circle theo-rem due to Jockusch, Propp and Shor [16], which describes the asymptoticbehavior of uniformly random domino tilings of the Aztec diamond. Roughly,the theorem states that the so-called polar regions , which are the four con-tiguous regions adjacent to the four corners of the Aztec diamond in whichthe tiling behaves in a predictable brickwork pattern, cover a region that isapproximately equal to the area that lies outside the circle inscribed in thediamond. See Figure 2, where the outline of the so-called “arctic” circle canbe clearly discerned. The precise statement is the following. Theorem 1 (The arctic circle theorem [16]).
Fix ε > . For each n ,consider a uniformly random domino tiling of AD n scaled by a factor /n in each axis to fit into the limiting diamondAD ∞ := {| x | + | y | ≤ } , RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX Fig. 2.
The arctic circle theorem: in a random domino tiling of AD , the circle-likeshape is clearly visible. Here, dominoes are colored according to their type and parity. and let P ◦ n ⊂ n − AD n be the image of the polar regions of the random tilingunder this scaling transformation. Then as n → ∞ the event that { ( x, y ) ∈ AD ∞ : x + y > + ε } ∩ ( n − AD n ) ⊂ P ◦ n ⊂ { ( x, y ) ∈ AD ∞ : x + y > − ε } holds with probability that tends to 1. In later work, Cohn, Elkies and Propp [2] derived more detailed asymp-totic information about the behavior of random domino tilings of the Aztecdiamond, that gives a quantitative description of the behavior of the tilinginside the arctic circle. They proved two main results (which are roughlyequivalent, if some technicalities are ignored), concerning the placementprobabilities (the probabilities to observe a given type of domino in a givenposition in the diamond) and the height function of the tiling (which, roughlyspeaking, encodes a weighted counting of the number of dominoes of differ-ent types encountered while travelling from a fixed place to a given positionin the diamond; see Section 6 for the precise definition).A main goal of this paper is to give a new proof of the Cohn–Elkies–Propp limit shape theorem for the height function of a uniformly randomdomino tiling of the Aztec diamond; see Theorem 12 in Section 6. Our proofis based on a large deviations analysis, and so gives some information that
D. ROMIK
Fig. 3.
A square Young tableau of order (shown in the “French” coordinate system)and the wall whose construction the tableau encodes at various stages of its construction. the proof in [2] (which is based on generating functions) does not: a largedeviation principle for the height function. Perhaps more importantly, ithighlights a surprising connection between the domino tilings model andanother, seemingly unrelated, combinatorial probability model, namely thatof random square Young tableaux .1.2. Random square Young tableaux.
Recall that a square (standard)Young tableau of order n is an array ( t i,j ) ni,j =1 of integers whose entriesconsist of the integers 1 , , . . . , n , each one appearing exactly once, andsuch that each row and column are arranged in increasing order. One canthink of a square Young tableau as encoding a sequence of instructions forconstructing an n × n wall of square bricks leaning against the x - and y -axesby laying bricks sequentially, where the rule is that each brick can be placedonly in a position which is supported from below and from the left by existingbricks or by the axes. In this interpretation, the number t i,j represents thetime at which a brick was added in position ( i, j ); see Figure 3. The numberof square Young tableaux of order n is known (via the hook-length formulaof Frame–Thrall–Robinson) to be ( n )! Q ni,j =1 ( i + j − . In [26], Boris Pittel and the author solved the problem of finding the limiting growth profile , or limit shape , of a randomly chosen square Youngtableau of high order. In other words, the question is to find the growthprofile of the square wall “constructed in the most random way.” Thiscan be expressed either in terms of the limit in probability L ( x, y ) of thescaled tableau entries n − t i,j , where ( x, y ) ∈ [0 , and i = i ( n ) and j = j ( n )are some sequences such that i/n → x and j/n → y as n → ∞ ; or alter-natively in terms of the limiting shape of the family of scaled “sublevelsets” { n − ( i, j ) : t i,j ≤ α · n } for each α ∈ (0 ,
1) (which in the “wall-building”metaphor represents the shape of the wall at various times, and thus can be
RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX Fig. 4.
The limiting growth profile of a random square Young tableau and the profile ofa randomly sampled tableau of order . The curves shown correspond to (scaled) times t = i/ , i = 1 , , . . . , . thought of as encoding the growth profile of the wall). Figure 4 shows thelimiting growth profile found by Pittel and Romik and the correspondingprofile of a randomly sampled square Young tableau of order 100.For the precise definition of the limiting growth profile, see [26]. Here, wemention only the following fact which will be needed in the next subsection:if L : [0 , × [0 , → [0 ,
1] is the limit shape function mentioned above, thenits values along the boundary of the square are given by L (0 , t ) = L ( t,
0) = 1 − √ − t ≤ t ≤ , (1) L (1 , t ) = L ( t,
1) = 1 + p t (2 − t )2 (0 ≤ t ≤ . (2)Also note that according to the limit shape theorem, the convergence of n − t i,j to L ( i/n, j/n ) as n → ∞ is uniform in i and j (this follows easilyfrom monotonicity considerations).1.3. An arctic circle theorem for square Young tableaux.
While it is notimmediately apparent from the description of this limit shape result, it fol-lows as a simple corollary of it that random square Young tableaux alsoexhibit an “arctic circle”-type phenomenon. That is, there is an equivalentway of visualizing the random tableau in which a spatial phase transitioncan be seen occurring along a circular boundary, where outside the circle thebehavior is asymptotically deterministic (the “frozen” phase) and inside thecircle the behavior is essentially random (the “disordered” or “temperate”phase). This fact, overlooked at the time of publication of the paper [26],was observed shortly afterwards by Benedek Valk´o [31]. In fact, deducingthe arctic circle result is easy and requires only the facts (1), (2) mentionedabove, which contain only a small part of the information of the limit shape.
D. ROMIK
To see how the arctic circle appears, we consider a different encoding ofthe information contained in the tableau via a system of particles on theinteger lattice Z . In this encoding we have n particles numbered 1 , , . . . , n ,where initially, each particle with index k is in position k . The particles areconstrained to remain in the interval [1 , n ]. At discrete time steps, particlesjump one step to the right, provided that the space to their right is empty(and provided that they do not leave the interval [1 , n ]). At each time step,exactly one particle jumps.It is easy to see that after exactly n steps, the system will terminate whenit reaches the state in which each particle k is in position n + k , and no furtherjumps can take place. We call the instructions for evolving the system ofparticles from start to finish a jump sequence . We can now add a probabilisticelement to this combinatorial model by considering the uniform probabilitymeasure on the set of all jump sequences of order n , and name the resultingprobability model the jump process of order n . But in fact, this is nothingmore than a thinly disguised version of the random square Young tableauxmodel, since jump sequences are in a simple bijection with square Youngtableaux: given a square tableau, think of the sequence of numbers in row k of the tableau as representing the sequence of times during which particle n + 1 − k jumps to the right. This is illustrated in Figure 5. We leave to thereader the easy verification that this gives the desired bijection.With these definitions, it is now natural to consider the asymptotic be-havior of this system of particles as n → ∞ . Figure 6 shows the result fora simulated system with n = 40. Here we see a circle-like shape appearingagain. To formulate precisely what is happening, given a jump process of Fig. 5.
The bijection between square Young tableaux and jump sequences: each row inthe tableau encodes the sequence of times at which a given particle jumps. As an example,the highlighted trajectory on the right-hand side corresponds to the highlighted row on theleft-hand side.
RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX Fig. 6.
A jump process with particles. order n , for each 1 ≤ k ≤ n , let τ − n ( k ) and τ + n ( k ) denote, respectively, thefirst and last times at which a particle k jumped from or to position k . Definethe frozen time-period in position k to be the union of the two intervals[0 , τ − n ( k )] ∪ [ τ + n ( k ) , n ] . Theorem 2 (The arctic circle theorem for random square Young tableaux).
Fix any ε > . Denote ϕ ± ( x ) = ± p x (1 − x ) . As n → ∞ , the event n max ≤ k ≤ n | n − τ − n ( k ) − ϕ − ( k/ n ) | < ε o ∩ n max ≤ k ≤ n | n − τ + n ( k ) − ϕ + ( k/ n ) | < ε o holds with probability that tends to . In other words, if the space–time di-agram of the trajectories in a random jump process is mapped to the unitsquare [0 , × [0 , by scaling the time axis by a factor /n and scaling theposition axis by a factor of / n , then for large n the frozen time-periodswill occupy approximately the part of the space–time diagram that lies in thecomplement of the disc { ( x, y ) ∈ R : ( x − / + ( y − / ≤ / } inscribed in the square. Proof.
First, note the following simple observations that express thetimes τ − n ( k ) and τ + n ( k ) in terms of the Young tableau ( t i,j ) ni,j =1 :(i) For 1 ≤ k ≤ n we have τ − n ( k ) = t n +1 − k, . D. ROMIK (ii) For n + 1 ≤ k ≤ n we have τ − n ( k ) = t ,k − n .(iii) For 1 ≤ k ≤ n we have τ + n ( k ) = t n,k .(iv) For n + 1 ≤ k ≤ n we have τ + n ( k ) = t n +1 − k,n .For example, the first statement is based on the fact that when 1 ≤ k ≤ n ,the time τ − n ( k ) is simply the first time at which the particle starting atposition k (which corresponds to row n + 1 − k in the tableau) jumps. Thethree remaining cases are equally simple and may be easily verified by thereader.Combining these observations with (1) and (2) and the limit shape the-orem, we now see that after scaling the times τ − n ( k ) and τ + n ( k ) by a factorof n − , we get quantities that converge in the limit, uniformly in k , to val-ues determined by the appropriate substitution of boundary values in thelimit shape function L ( x, y ). For example, to deal with case (i) above, when1 ≤ k ≤ n , using (1) we have that n − τ − n ( k ) = n − t n +1 − k, ≈ L (cid:18) , − k − n (cid:19) = 1 − p − (1 − ( k − /n ) − p ( k − /n (1 − ( k − /n )2= ϕ − (cid:18) k − n (cid:19) ≈ ϕ − ( k/ n ) , uniformly in 1 ≤ k ≤ n . Similarly, the other three cases each imply that n − τ ± n ( k ) is uniformly close to ϕ ± ( k/ n ) in the appropriate range of valuesof k ; we omit the details. Combining these four cases gives exactly that theevent in Theorem 2 holds with asymptotically high probability as n → ∞ . (cid:3) Similarity of the models and the analytic technique.
Apart from giv-ing a new proof of the limit shape theorem of Cohn, Elkies and Propp, an-other main goal of this paper is to show that the two models described inthe preceding sections (random domino tilings of the Aztec diamond andrandom square Young tableaux) exhibit similar behavior on a more detailedlevel than that of the mere appearance of the arctic circle, and that in factthey are almost equivalent in an asymptotic sense. Our new proof of thelimit shape theorem for the height function will use the same techniquesdeveloped in [26] for the case of random square Young tableaux: we firstderive a large deviations principle, not for domino tilings but for a relatedmodel of random alternating sign matrices , then solve the resulting problemin the calculus of variations using an analysis that parallels, to a remarkable(and, in our opinion, rather surprising) level of similarity, the analysis ofthe variational problem in [26]. The resulting formulas for the solution of
RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX the variational problem are almost identical to the formulas for the limitinggrowth profile of random square Young tableaux. Up to some trivial scalingfactors related to the choice of coordinate system, the formulas for the twolimit shapes can be written in such a way that the only difference betweenthem is a single minus sign.Another important aspect of our results lies not in the results themselvesbut in the techniques used. We use the methods first presented in [26] tosolve another variational problem belonging to a class of problems previ-ously thought to be difficult to analyze, due to a lack of a systematic frame-work that enables one to derive the solution in a relatively mechanical way(as opposed to having to guess it using some deep analytic insight) andthen rigorously verify its claimed extremal properties. This justifies to someextent the claim from [26] that the analytic techniques of that paper pro-vide a systematic approach for dealing with such problems, which seem toappear frequently in the analysis of combinatorial probability models (see[4, 23, 26, 32, 33]), and are also strongly related to classical variationalproblems arising in electrostatics and in random matrix theory.The rest of the paper is organized as follows. In Section 2 we recall somefacts about alternating sign matrices, and study the problem of finding thelimiting height matrix of an alternating sign matrix chosen randomly ac-cording to domino measure , which is a natural (nonuniform) probabilitymeasure on the set of alternating sign matrices of order n . In Section 3 wederive a large deviation principle for this model. This problem is solved inSection 4. In Section 5 we prove a limiting shape theorem for the heightmatrix of an alternating sign matrix chosen according to domino measure.In Section 6 we deduce from the previous results the Cohn–Elkies–Propplimiting shape theorem for the height function of uniformly random dominotilings of the Aztec diamond. Section 7 has some final remarks, includinga discussion on the potential applicability of our methods to attack the well-known open problem of the limit shape of uniformly random alternating signmatrices.
2. Alternating sign matrices. An alternating sign matrix (often abbre-viated as ASM ) of order n is an n × n matrix with entries in { , − , } such that in every row and every column, the sum of the entries is 1, andthe nonzero numbers appear with alternating signs. See Figure 7(a) for anexample. Alternating sign matrices were first defined and studied in theearly 1980s by Robbins and Rumsey in connection with their study [29] ofDodgson’s condensation method for computing determinants and of the λ -determinant , a natural generalization of the determinant that arises fromthe condensation algorithm. Later, Robbins, Rumsey and Mills publishedseveral intriguing theorems and conjectures about them [24], tying them to D. ROMIK − − − (a) (b) Fig. 7. (a)
An ASM of order 6; (b) its height matrix. the study of plane partitions and leading to many later interesting develop-ments, some of which are described, for example, in [1, 28].Denote by A n the set of ASMs of order n . For a matrix M ∈ A n , denoteby N + ( M ) the number of its entries equal to 1. An important formula provedby Mills, Robbins and Rumsey states that X M ∈A n N + ( M ) = 2( n +12 ) . (3)This is sometimes referred to as the “2-enumeration” of ASMs. The readermay note that the right-hand side is equal to the number of domino tilings ofAD n mentioned at the beginning of the Introduction; indeed, a combinatorialexplanation for (3) in terms of domino tilings was found by Elkies et al. [6].In Section 6 we will say more about this connection and how to make useof it, but for now, we rewrite (3) more probabilistically as2 − ( n +12 ) X M ∈A n N + ( M ) = 1 , and consider this as the basis for defining a probability measure on A n ,which we call domino measure (thus named since it is closely related to theuniform measure on domino tilings of AD n ; see Section 6), given by theexpression P n Dom ( M ) = 2 N + ( M ) − ( n +12 ) ( M ∈ A n ) . Our first goal will be to study the asymptotic behavior of large randomASMs chosen according to domino measure, and specifically the limit shapeof their height matrix . The height matrix of an ASM M = ( m i,j ) ni,j =1 ∈ A n is defined to be the new matrix H ( M ) = ( h i,j ) ni,j =0 of order ( n + 1) × ( n + 1)whose entries are given by h i,j = X p ≤ i X q ≤ j m p,q . RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX The matrix H ( M ) is also sometimes referred to as the corner sum matrix of M . It satisfies the following conditions: h ,k = h k, = 0 for all 0 ≤ k ≤ n, (H1) h n,k = h k,n = k for all 0 ≤ k ≤ n, (H2) 0 ≤ h i +1 ,j − h i,j , h j,i +1 − h j,i ≤ ≤ i < n, ≤ j ≤ n. (H3)See Figure 7(b) for an example. (In fact, it is not too difficult to see thatthe correspondence M → H ( M ) defines a bijection between the set of ASMsof order n and the set of matrices satisfying conditions (H1)–(H3) (see [29],Lemma 1) but we will not need this fact here.) In particular, the “Lipschitz”-type condition (H3) means that the height matrix can be thought of asa discrete version of a two-dimensional surface, and is therefore a naturalcandidate for which to try and prove a limit shape result.The basis for our analysis of P n Dom -random ASMs is a formula which willgive the probability distribution (under the measure P n Dom ) of the k th rowof the height matrix, for each 1 ≤ k ≤ n . To describe this, first, as usual,denote the Vandermonde function by∆( u , . . . , u m ) = Y ≤ i If integers ≤ x < x < · · · < x k ≤ n are given, and if y < y < · · · < y n − k are the numbers in { , , . . . , n } \ { x , . . . , x k } arrangedin increasing order, then, in the notation above, we have P n Dom [ M ∈ A n : ( X k (1) , . . . , X k ( k )) = ( x , . . . , x k )](4) = 2( k +12 )2( n − k +12 )2( n +12 ) · ∆( x , . . . , x k )∆( y , . . . , y n − k )∆(1 , , . . . , k )∆(1 , , . . . , n − k ) . To prove Theorem 3, we use another well-known combinatorial bijectionrelating ASMs to monotone triangles . A monotone triangle of order n isa triangular array ( t i,j ) ≤ i ≤ n, ≤ j ≤ i of integers satisfying the inequalities t i,j < t i,j +1 , t i,j ≤ t i − ,j ≤ t i,j +1 (2 ≤ i ≤ n, ≤ j ≤ i − . D. ROMIK 32 51 4 51 2 4 61 2 3 4 61 2 3 4 5 6 1 2 3 4 5 61 2 4 5 61 3 4 62 3 63 55(a) (b) Fig. 8. (a) The complete monotone triangle corresponding to the ASM in Figure 7; (b) itsdual, shown “standing on its head.” A complete monotone triangle of order n is a monotone triangle whose bot-tom row consists of the numbers (1 , , . . . , n ). It is well known that alter-nating sign matrices of order n are in bijection with complete monotonetriangles of order n . In our terminology, the bijection assigns to an ASM M = ( m i,j ) ni,j =1 the monotone triangle T = ( t i,j ) ≤ i ≤ n, ≤ j ≤ i = ϕ ASM → CMT ( M )whose k th row ( t k,j ) ≤ j ≤ k consists for each 1 ≤ k ≤ n of the ascents of the k throw of the height matrix H ( M ), arranged in increasing order. See Figure 8(a)for an example. More explicitly, it is easy to check that this means that anindex j will be present in the k th row of T if and only if k X i =1 m i,j = 1holds.Another notion that will prove useful is that of the dual of a completemonotone triangle. If T is a complete monotone triangle of order n , and M is the ASM in A n such that T = ϕ ASM → CMT ( M ), then the dual T ∗ of T isthe complete monotone triangle of order n that corresponds via the samebijection to the matrix W , defined as the vertical reflection of M , that is,the matrix such that w i,j = m n +1 − i,j for all i, j (clearly it, too, is an ASM).See Figure 8(b), where the dual triangle is drawn reflected vertically.The following simple observation describes more explicitly the connectionbetween a monotone triangle and its dual. Lemma 4. If T = ( t i,j ) ≤ i ≤ n, ≤ j ≤ i is a complete monotone triangle oforder n , then for each ≤ k ≤ n − , the ( n − k ) th row of the dual triangle T ∗ consists of the numbers in the complement { , , . . . , n } \ { t k, , t k, , . . . , t k,k } of the k th row of T , arranged in increasing order. RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX Proof. Let M = ( m i,j ) i,j ∈ A n be such that T = ϕ ASM → CMT ( M ). Asmentioned above, 1 ≤ j ≤ n appears in the k th row of T if and only if P ki =1 m i,j = 1. Similarly, from the definition of T ∗ we see that j appearsin the ( n − k )th row of T ∗ if and only if P ni = k +1 m i,j = 1. But from thedefinition of an alternating sign matrix, one and only one of these conditionsmust hold. (cid:3) As the last step in the preparation for proving Theorem 3, we note thatif M ∈ A n and T = ϕ ASM → CMT ( M ), then it is easy to see that N + ( M ), thenumber of +1 entries in M , can be expressed in terms of T as the num-ber of entries t i,j in T that do not appear in the preceding row (including,vacuously, the singleton element in the top row). We denote this quantityalso by N + ( T ); note that it is defined more generally also for noncompletemonotone triangles. We furthermore recall the following formula proved byMills, Robbins and Rumsey in [24], Theorem 2, (see also [6], equation (7),Section 4, and see [12] for a recent alternative proof and some generaliza-tions): Lemma 5. If k ≥ and x < x < · · · < x k are integers, then the sumof N + ( T ) over all monotone triangles T of order k with bottom row ( x , . . . ,x k ) is equal to k +12 ) Y ≤ i Denote by T n ( x , . . . , x k ) the set of completemonotone triangles of order n whose k th row is equal to ( x , . . . , x k ). Fromthe remarks above, it follows that the left-hand side of (4) is equal to2 − ( n +12 ) X T ∈T n ( x ,...,x k ) N + ( T ) . In addition, for a monotone triangle T ∈ T n ( x , . . . , x k ), define T top and T bottom as the two monotone triangles, of orders k and n − k , respectively, where T top is comprised of the top k rows of T , and T bottom is comprised of the top n − k rows of the dual triangle T ∗ . From Lemma 4, it follows that the correspon-dence T → ( T top , T bottom )defines a bijection between T n ( x , . . . , x k ) and the cartesian product A × B ,where A is the set of monotone triangles with bottom row ( x , . . . , x k ) and B is the set of monotone triangles with bottom row ( y , . . . , y n − k ) (in the no-tation of Theorem 3). This correspondence furthermore has the propertythat N + ( T ) = N + ( T top ) + N + ( T bottom ) D. ROMIK [since N + ( T top ) counts the number of +1 entries in the first k rows of theASM corresponding to T , whereas N + ( T bottom ) counts the number +1’sin the last n − k rows], or equivalently that 2 N + ( T ) = 2 N + ( T top ) N + ( T bottom ) .Combining these last observations, we get that the left-hand side of (4) isequal to 2 − ( n +12 ) X T top ∈A N + ( T top ) X T bottom ∈B N + ( T bottom ) , which by Lemma 5 is equal exactly to the right-hand side of (4). (cid:3) We remark that an equivalent version of Theorem 3, phrased in the lan-guage of domino tilings and certain so-called zig–zag paths defined in termsof them, is proved by Johansson in [17] [see Proposition 5.14 in that paperand equation (5.16) following it]. See also the subsequent papers [18, 19]where Johansson proves many interesting results about random dominotilings of the Aztec diamond by combining a variant of (4) with ideas fromthe theory of orthogonal polynomials and the theory of determinantal pointprocesses. 3. A large deviation principle. We now turn from combinatorics to anal-ysis, with the goal in mind being to use Theorem 3 as the starting point fora large deviation analysis of the behavior of P n Dom -random ASMs. First, wedefine the space of functions on which our analysis takes place. Fix 0 < y < k th row of the height matrix ofa P n Dom -random ASM of order n for values of k satisfying k ≈ y · n , when n is large.Define the space of y -admissible functions to be the set F y = { f : [0 , → [0 , 1] : f is monotone nondecreasing, 1-Lipschitz,and satisfies f (0) = 0 , f (1) = y } . Define the space of admissible functions as the union of all the y -admissiblefunction spaces: F = [ y ∈ [0 , F y . We also define a discrete analogue of the admissible functions. Givenintegers 0 ≤ k ≤ n , a sequence u = ( u , u , . . . , u n ) of integers is called an( n, k ) -admissible sequence if it satisfies u = 0 , u n = k and u i +1 − u i ∈ { , } for all 0 ≤ i ≤ n − . Note that ( n, k )-admissible sequences are exactly those that can appear asthe k th row of a height matrix H ( M ) of an ASM M ∈ A n . We embed the RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX Fig. 9. A (6 , -admissible sequence u and the corresponding function f u . ( n, k )-admissible sequences in the space F y for y = k/n , in the following way:for each ( n, k )-admissible sequence u , define a function f u : [0 , → [0 , 1] asthe unique function having the values f u ( j/n ) = u j /n, ≤ j ≤ n, and on each interval [ j/n, ( j + 1) /n ] for 0 ≤ j ≤ n − f u is a ( k/n )-admissible function. In fact, it is easy to seethat the admissible functions are precisely the limits of such functions in theuniform norm topology.With these definitions, we can now formulate the large deviation principle. Theorem 6 (Large deviation principle for P n Dom -random ASMs). Let ≤ k ≤ n , and let u = ( u , u , . . . , u n ) be an ( n, k ) -admissible sequence. Let H ( M ) k denote the k th row of a height matrix H ( M ) . Then P n Dom [ M ∈ A n : H ( M ) k = u ](5) = exp( − (1 + o (1)) n ( I ( f u ) + θ ( k/n ))) , where we define θ ( y ) = 12 y log y + 12 (1 − y ) log(1 − y ) + 2 log 2 − y (1 − y ) + 32 ,I ( f ) = − Z Z log | s − t | f ′ ( s )( f ′ ( t ) − ds dt ( f ∈ F ) . The o (1) error term in (5) is uniform over all ≤ k ≤ n and all ( n, k ) -admissible sequences u , as n → ∞ . Proof. Let 1 ≤ x < x < · · · < x k ≤ n be the positions of the k ascentsin the sequence ( u , u , . . . , u n ) (in the same sense defined before, namelythat u x i − u x i − = 1), and let 1 ≤ y < · · · < y n − k ≤ n be the numbers in thecomplement { , . . . , n } \ { x , . . . , x k } arranged in increasing order. D. ROMIK By (4), we have n − log P n Dom [ M ∈ A n : H ( M ) k = u ]= n − (cid:18)(cid:18) k + 12 (cid:19) + (cid:18) n − k + 12 (cid:19) − (cid:18) n + 12 (cid:19)(cid:19) log 2(6) − n − X ≤ i 12 ( X + 1) log (cid:18) X − X (cid:19) + X log (cid:18) X + 1 X − (cid:19) − (cid:19) . When X is large, this behaves like log X + O ( X ). The integral is also definedand finite when X = 1. So we can write n − X ≤ i 1] : s < t } , at the cost of an additional error which can be bounded in absolute valueby Z dy Z yy − /n | log( y − x ) | dx = (cid:12)(cid:12)(cid:12)(cid:12)Z /n log t dt (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) log nn (cid:19) . D. ROMIK To summarize, after this change in the region of integration and, in addition,after symmetrizing the region of integration for convenience, we have shownthat n − X ≤ i 34 + 34 y + 34 (1 − y ) − log 2 · y (1 − y ) − y log y − 12 (1 − y ) log(1 − y )+ Z Z log | t − s | f ′ u ( s ) f ′ u ( t ) ds dt (13) − Z Z log | t − s | f ′ u ( s ) ds dt + o (1)= − θ ( y ) − I ( f u ) + o (1)as claimed. (cid:3) 4. The variational problem and its solution. Fix 0 < y < 1. Motivatedby Theorem 6, we now turn our attention to the problem of minimizing theintegral functional I ( f ) over the appropriate class of y -admissible functions.In the next section we will show how this implies a limit shape result for P n Dom -random ASMs. RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX The precise variational problem that we will solve is the following: Variational Problem 1. For a given < y < , find the function f ∗ y that minimizes I ( f ) over all functions f ∈ F y . Variational Problem 1 is a variant of a class of variational problems thathave appeared in several random combinatorial models (see, e.g., [4, 23,26, 32, 33]). Such problems bear a strong resemblance to classical physi-cal problems of finding the distribution of electrostatic charges subject tovarious constraints in a one-dimensional space, as well as to problems of find-ing limiting eigenvalue distributions in random matrix theory. However, thevariational problems arising from combinatorial models usually have non-physical constraints that make the analysis trickier. In particular, in severalof the works cited above, the presence of such constraints required the au-thors to first (rather ingeniously) guess the solution. Once the solution wasconjectured, it was possible to verify that it is indeed the correct one usingfairly standard techniques. Cohn, Larsen and Propp, who derived the limitshape of a random boxed plane partition, ask (see Open Question 6.3 in [4])whether there exists a method of solution for their problem that does notrequire guessing the solution.In [26], it was argued, however, that when dealing with such problems,it is not necessary to guess the solution, since a well-known formula in thetheory of singular integral equations for inverting a Hilbert transform ona finite interval actually enables mechanically deriving the solution ratherthan guessing it, once certain intuitively plausible assumptions on the formof the solution are made. Here, we demonstrate again the use of this moresystematic approach by using it to solve our variational problem. As anadded bonus, the solution rather elegantly turns out to be nearly identicalto the solution of the variational problem for the square Young tableauxcase (although we see no a priori reasons why this should turn out to be thecase), and we are able to make use of certain nontrivial computations thatappeared in [26], which further simplifies the analysis.Our goal in the rest of this section will be to prove the following theorem. Theorem 7. Define Z ( x, y ) = 2 π (cid:20) ( x − / 2) arctan (cid:18) p / − ( x − / − ( y − / / − y (cid:19) + 12 arctan (cid:18) x − / / − y ) p / − ( x − / − ( y − / (cid:19) (14) − (1 / − y ) arctan (cid:18) x − / p / − ( x − / − ( y − / (cid:19)(cid:21) . D. ROMIK For < y < / , the solution f ∗ y to Variational Problem 1 is given by f ∗ y ( x ) = , ≤ x ≤ − p y (1 − y )2 , y Z ( x, y ) , − p y (1 − y )2 < x < p y (1 − y )2 , y, p y (1 − y )2 ≤ x ≤ . (15) For y = 1 / , the solution is given by f ∗ / ( x ) = x . For y > / the solution is expressed in terms of the solution for − y by f ∗ y = x − f ∗ − y . Moreover, for all < y < we have I ( f ∗ y ) = − θ ( y ) . As a first step, for convenience we reformulate the variational problemslightly to bring it to a more symmetric form, by replacing each f ∈ F y bythe function g ( x ) = 2 f ( x ) − x. (16)It is easy to check how the class of y -admissible functions and the func-tional I ( · ) transform under this mapping. The result is the following equiv-alent form of our variational problem. Variational Problem 2. For < y < , define the space of functions G y = { g : [0 , → [ − , 1] : g (0) = 0 , g (1) = 2 y − , and g is 1-Lipschitz } and the integral functional J ( g ) = − Z Z g ′ ( s ) g ′ ( t ) log | s − t | ds dt. Find the function g ∗ y ∈ G y that minimizes the functional J over all functions g ∈ G y . The reader may verify that if f ∈ F y and g ∈ G y are related by (16), thenthe integral functionals I and J are related by I ( f ) = J ( g ) − . This implies that the following theorem is an equivalent version of Theo-rem 7. RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX Theorem 7 ′ . For < y < / , the solution g ∗ y to Variational Problem 2is given by g ∗ y ( x ) = − x, ≤ x ≤ − p y (1 − y )2 , y − x + Z ( x, y ) , − p y (1 − y )2 < x < p y (1 − y )2 , y − x, p y (1 − y )2 ≤ x ≤ ,where Z ( x, y ) is defined in (14). For y = 1 / , the solution is given by g ∗ / ( x ) ≡ . For y > / the solution is expressed in terms of the solution for − y by g ∗ y = − g ∗ − y . Moreover, for all < y < we have J ( g ∗ y ) = − θ ( y ) + . Proof. We now concentrate our efforts on proving Theorem 7 ′ . First,in the following lemma we recall some basic facts about the space G y andthe functional J . We omit the proofs, since they are relatively simple andessentially the same claims, with minor differences in the coordinate system,were proved in [26]. (See also [4] where similar facts are proved.) Lemma 8. (i) The space G y is compact in the uniform norm. (ii) The functional J on G = S 2, where clearly g ∗ / ≡ J among all Lipschitz functions, and in partic-ular on G / . It is also easy to see that a function g is the minimizer for J on G y if and only if − g is the minimizer on G − y . So we may assume for therest of the discussion that y < / D. ROMIK With these preparations, we can start the analysis. We need to minimi-ze J ( g ) under the constraints g ∈ G y , which we rewrite as:(i) g (0) = 0;(ii) g is differentiable almost everywhere and g ′ satisfies − ≤ g ′ ≤ R g ′ ( x ) dx = 2 y − J as being defined on the largerspace G and form the Lagrangian L ( g, λ ) = J ( g ) − λ Z g ′ ( x ) dx, where λ is a Lagrange multiplier. Minimizing J under this constraint leads,via the usual recipe for constrained optimization, to the equation W ( s ) := − Z g ′ ( t ) log | s − t | dt − λ = 0 . (18)The reason for this is that, informally, W ( s ) as defined above can be thoughtof as “the partial derivative of L with respect to g ′ ( s )” [where we thinkof L as a function of the uncountably many variables ( g ′ ( s )) s ∈ [0 , , which isa standard point of view in the variational calculus].Relation (18) should hold whenever g ′ ( s ) is defined and is in ( − , g ′ = − g ′ = 1. The correct condition (the so-called “complementary slackness”condition) is given by the following lemma. Lemma 9. If g ∈ G y and for some real number λ the function W ( s ) defined in (18) satisfies W ( s ) is = 0 , if g ′ ( s ) ∈ ( − , , ≥ , if g ′ ( s ) = − , ≤ , if g ′ ( s ) = 1 , (19) then g = g ∗ y is the minimizer for J in G y . Proof. We copy the proof almost verbatim from [26], Lemma 7. If h ∈ G y , then in particular h is 1-Lipschitz, so( h ′ ( s ) − g ′ ( s )) W ( s ) ≥ s for which this is defined. So Z h ′ ( s ) W ( s ) ds ≥ Z g ′ ( s ) W ( s ) ds or in other words 2 h g, h i − λ (2 y − ≥ h g, g i − λ (2 y − , RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX which shows that h g, h i ≥ h g, g i . Therefore we get, using Lemma 8(ii), that h h, h i = h g, g i + 2 h g, h − g i + h h − g, h − g i ≥ h g, g i as claimed. (cid:3) Having established a sufficient condition [comprised of the three separateconditions in (19)] for a function to be a minimizer, we first try to satisfycondition (18) and save the other conditions for later. Based on intuitionthat comes from the problem’s connection to the combinatorial model, wemake the assumption that the minimizer g is piecewise smooth and satisfies g ′ ( s ) ∈ ( − , 1) if s ∈ (cid:20) − β , β (cid:21) , (20) g ′ ( s ) = − s / ∈ (cid:20) − β , β (cid:21) , (21)where β = 2 p y (1 − y ) . Note that g ′ ( s ) = − f ′ ( s ) = 0 in the original space F y of y -admissible functions, which corresponds to having no ascents (or veryfew ascents) in the vicinity of the scaled position ( s, y ) in the height matrix ofthe ASM. Our knowledge of the endpoints of the interval in which g ′ ( s ) > − g into (18) gives the equation − Z (1+ β ) / − β ) / g ′ ( t ) log | s − t | dt = 12 λ − s log s − (1 − s ) log(1 − s ) + (cid:18) s − − β (cid:19) log (cid:18) s − − β (cid:19) + (cid:18) β − s (cid:19) log (cid:18) β − s (cid:19) − β, s ∈ (cid:18) − β , β (cid:19) . Differentiating with respect to s then gives − Z (1+ β ) / − β ) / g ′ ( t ) s − t dt = − log s + log(1 − s )(22) + log (cid:18) s − − β (cid:19) − log (cid:18) β − s (cid:19) . D. ROMIK So, just like in the analysis in [26], we have reached the problem of invert-ing a Hilbert transform on a finite interval (the so-called airfoil equation ).Moreover, the function whose inverse Hilbert transform we want to computeis very similar to the one that appeared in [26]—in fact, up to scaling factorsonly the signs of some of the terms are permuted, and in [26] there is anextra term equal to the Lagrange multiplier λ .Now recall that in fact the general form of the solution of equations of thistype is known. The following theorem appears in [7], Section 3.2, page 74(see also [27], Section 9.5.2): Theorem 10. The general solution of the airfoil equation π Z − h ( u ) u − v du = p ( v ) , | v | < , with the integral understood in the principal value sense, and h satisfyinga H¨older condition, is given by h ( v ) = 1 π √ − v Z − √ − u p ( u ) v − u du + c √ − v for some c . Now set h ( v ) = g ′ ((1 + βv ) / . (23)This function should satisfy Z − h ( u ) u − v du = log (cid:18) − βu (cid:19) − log (cid:18) βu (cid:19) + log(1 + u ) − log(1 − u ) , so, applying Theorem 10, we get the equation h ( v ) = 1 π √ − v Z − √ − u v − u (cid:20) log (cid:18) u − u (cid:19) + log (cid:18) − βu βu (cid:19)(cid:21) du + c √ − v , where c is an arbitrary constant. This can be written as h ( v ) = 1 π √ − v ( I ( v, /β ) + I ( − v, /β )) + c √ − v , (24)where I is defined by I ( ξ, γ ) = Z − p − η ξ − η log (cid:18) ηγ + η (cid:19) dη RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX and is evaluated in [26], Lemma 8, as I ( ξ, γ ) = π " − γ + p γ − − ξ arccosh( γ ) − p − ξ arctan s ( γ − − ξ )( γ + 1)(1 + ξ ) . Therefore we get that h ( v ) = 1 π √ − v (cid:18) c + β − p − β β (cid:19) − π arctan s ( β − − − v )( β − + 1)(1 + v ) + arctan s ( β − − v )( β − + 1)(1 − v ) ! . Since c is an arbitrary constant, we see that the only sensible choice thatwill allow h to be a bounded function on the interval ( − , 1) is that of c = − ( β − p − β ) /β . So we have h ( v ) = − π arctan s ( β − − − v )( β − + 1)(1 + v ) + arctan s ( β − − v )( β − + 1)(1 − v ) ! . At this point, it is worth pointing out that in (24), if we had the difference of the two I integrals instead of their sum, we would get at the end (up tosome trivial scaling factors that are due to the use of different coordinatesystems) exactly the function from the paper [26] that solves the variationalproblem for random square Young tableaux! (Compare with equation (36)in [26] and subsequent formulas.) Thus, while the variational problems aris-ing from these two combinatorial models are not exactly isomorphic (whichwould be perhaps less surprising), they are in some sense nearly equivalent.It would be interesting to understand if this phenomenon has a conceptualexplanation of some sort, but we do not see one at present.Simplifying the expression for h using the sum-of-arctangents identityarctan X + arctan Y = arctan X + Y − XY gives h ( v ) = − π arctan s − β β − β v . Going back to the original function g related to h via (23), we get that g ′ ( s ) = h ((2 s − /β ) = − π arctan s / − y (1 − y ) s (1 − s ) + y (1 − y ) − / D. ROMIK = − π arctan (cid:18) / − y p / − ( y − / − ( s − / (cid:19) = 2 π arctan (cid:18) p / − ( y − / − ( s − / / − y (cid:19) − s ∈ ( − β , β ). From this, we can now get g by integration. First, from (21)we obtain that g ( s ) = − s if 0 ≤ s ≤ − β . Next, in the interval ( − β , β ) we can integrate g ′ using the identity Z t arctan p a − u du = t arctan p a − t + √ a arctan (cid:18) t √ a √ a − t (cid:19) − arctan (cid:18) t √ a − t (cid:19) ( t < a ) , and obtain without much difficulty that g ( s ) = g (cid:18) − β (cid:19) + Z s (1 − β ) / g ′ ( x ) dx = y − s + 2 π (cid:20) ( s − / 2) arctan (cid:18) p / − ( s − / − ( y − / / − y (cid:19) + 12 arctan (cid:18) s − / / − y ) p / − ( s − / − ( y − / (cid:19) − (1 / − y ) arctan (cid:18) s − / p / − ( s − / − ( y − / (cid:19)(cid:21) for s ∈ ( − β , β ).Finally, from this last equation it is easy to check that g (cid:18) β (cid:19) = lim s ↑ (1+ β ) / g ( s ) = 2 y − β , so, for s > β , again because of (21) we get that g ( s ) = 2 y − s . In particu-lar, g satisfies the conditions g (0) = 0 , g (1) = 2 y − 1, and it is also 1-Lipschitz,so g ∈ G y .To summarize, we have recovered as a candidate minimizer exactly thefunction from Theorem 7 ′ . We also verified that it is in G y . Furthermore, bythe derivation and the use of Theorem 10, we know that it satisfies (22), or in RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX other words that W ′ ( s ) ≡ − β , β ). We wanted to show that W ( s ) ≡ W ( s ) in (18), we see that weare still free to choose the Lagrange multiplier λ , which starting from (22)has disappeared from the analysis! So, taking λ = − R g ′ ( t ) log | t − / | dt ensures that (18) holds on ( − β , β ), which is one of the sufficient conditionsin Lemma 9.All that remains to finish the proof that g = g ∗ y is the minimizer is toverify the second and third conditions in (19), which we have not considereduntil now. The third condition is irrelevant, since g ′ is never equal to 1, sowe need to prove that W ( s ), which we will now re-denote by W ( s, y ) toemphasize its dependence on y , is nonnegative when s / ∈ [ − β , β ]. Since g ′ is an even function, it follows that W ( · , y ) is also even, so it is enough tocheck this when s > β .Once again, our argument follows closely in the footsteps of the analogouspart of the proof in [26]. Fix 1 / < s ≤ 1, and let ˆ y = −√ − s , so that β (ˆ y ) = s . We know from (18) that W ( s, ˆ y ) = 0. To finish the proof, it isenough to show that ∂W ( s, y ) ∂y ≤ ≤ y ≤ ˆ y. Denote G ( x, y ) = g ∗ y ( x ). Then ∂W ( s, y ) ∂y = − Z ∂ G ( t, y ) ∂t ∂y log | s − t | dt + 2 Z ∂ G ( t, y ) ∂t ∂y log | t − / | dt. A computation shows that if t ∈ ( − β ( y )2 , β ( y )2 ) then ∂ G ( t, y ) ∂t ∂y = ∂∂y g ∗ y ′ ( x ) = 2 π · p / − ( x − / − ( y − / , and otherwise ∂ G ( t, y ) /∂t ∂y is clearly 0, so that ∂W ( s, y ) ∂y = 4 π Z (1+ β ) / − β ) / log | t − / | − log( s − t ) p / − ( t − / − ( y − / dt. Now use the two standard integral evaluations Z − log | x |√ − x dx = − π log(2) , Z − log( a − x ) √ − x dx = π log (cid:18) a + √ a − (cid:19) ( a > ∂W ( s, y ) ∂y = − (cid:18) s − / p ( s − / − ( β/ β/ (cid:19) . D. ROMIK Since we assumed that y ≤ ˆ y , or in other words that s ≥ β ( y )2 , it followsthat ∂W ( s, y ) ∂y ≤ − (cid:18) s − / β/ (cid:19) ≤ ′ (hence also Theorem 7), except the claimabout the value of the integral functional J at the minimizer g ∗ y . This valuecould be computed in a relatively straightforward way, as was done for theanalogous claim in [26]. We omit this computation, since, as was pointedout in [26], this can also be proved indirectly by using the large deviationprinciple to conclude that the infimum of the large deviations rate functional I ( f ) + θ ( y ) over the space F y must be equal to 0. Therefore the proof ofTheorem 7 ′ is complete. (cid:3) 5. The limit shape of P n Dom -random ASMs. We now apply the resultsfrom the previous sections to prove a limit shape result for the height matrixof random ASMs chosen according to the measure P n Dom . Theorem 11. Let F ( x, y ) = f ∗ y ( x ) , where for each ≤ y ≤ , f ∗ y is thefunction defined in (15). For each n let M n be a P n Dom -random ASM oforder n , and let H n = H ( M n ) = ( h ni,j ) ni,j =0 be its associated height matrix.Then as n → ∞ we have the convergence in probability max ≤ i,j ≤ n (cid:12)(cid:12)(cid:12)(cid:12) h ni,j n − F ( i/n, j/n ) (cid:12)(cid:12)(cid:12)(cid:12) P −→ n →∞ . Proof. Fix ε > 0. We want to show that A nε = (cid:26) max ≤ i,j ≤ n (cid:12)(cid:12)(cid:12)(cid:12) h ni,j n − F ( i/n, j/n ) (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:27) satisfies P n Dom ( A nε ) → n → ∞ . We start by showing a weaker statement,namely that if y ∈ (0 , 1) is given, then P n Dom ( B nε,y ) → n → ∞ , where B nε,y = (cid:26) max ≤ j ≤ n (cid:12)(cid:12)(cid:12)(cid:12) h n ⌊ ny ⌋ ,j n − F ( y, j/n ) (cid:12)(cid:12)(cid:12)(cid:12) > ε/ (cid:27) (and ⌊ x ⌋ denotes as usual the integer part of a real number x ). To provethis, note that B nε,y ⊆ [ u { M ∈ A n : H ( M ) ⌊ ny ⌋ = u } , where the union is over all ( n, k )-admissible sequences u (with k = ⌊ ny ⌋ )such that k f u − f ∗ y k ∞ = max ≤ x ≤ | f u ( x ) − f ∗ y ( x ) | > ε/ RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX (here, k · k ∞ denotes the supremum norm on continuous functions on [0 , n, k )-admissible sequences, which is equal to (cid:0) nk (cid:1) ≤ n [since an ( n, k )-admissiblesequence is determined by the positions of its k ascents], and for each such u ,by Theorem 6 we have P n Dom ( M ∈ A n : H ( M ) ⌊ ny ⌋ = u ) ≤ C exp( − (1 + o (1)) c ( ε, y ) n ) , where C is a universal constant, and c ( ε, y ) = inf { I ( f ) + θ ( y ) : f ∈ F y , k f − f ∗ y k ∞ ≥ ε/ } . (25)If the infimum in the definition of c ( ε, y ) were taken over all f ∈ F y , itwould be equal to 0 by Theorem 7. Note, however, that the set of g ∈ G y that correspond via (16) to some f ∈ F y participating in the infimum in (25)is a closed subset (in the uniform norm topology) of G y that does not con-tain the minimizer g ∗ y . Therefore by Theorem 7 ′ and Lemma 8 we get thatin fact c ( ε, y ) > 0. Combining these last observations, we see that indeed P n Dom ( B nε,y ) → n → ∞ .Next, we claim that the event A nε is contained in the union of a finitenumber (that depends on ε but not on n ) of events B nε,y j , so if P n Dom ( B nε,y ) → y then also P n Dom ( A nε ) → 0. This follows because of the Lipschitzproperty of the height matrix and of the limit shape function F , which meansthat proximity to the limit at a sufficiently dense set of values of y impliesproximity to the limit everywhere. The details are simple, so we leave tothe reader to check that taking y j = ⌊ jε/ ⌋ for j = 1 , , . . . , ⌊ /ε ⌋ is in factsufficient to guarantee that A nε ⊂ ⌊ /ε ⌋ [ j =1 B nε,y j as required. (cid:3) In the next section we will use a connection between uniformly randomdomino tilings of the Aztec diamond and P n Dom -random ASMs to provea limit shape theorem for the height function of the random domino tiling.It will be helpful to consider for this purpose a variant of the height matrixof an ASM M , which we call the symmetrized height matrix (it is sometimesreferred to as the skewed summation of M ). If M ∈ A n , we define this asthe matrix H Sym ( M ) = ( h ∗ i,j ) ni,j =0 with entries given by h ∗ i,j = i + j − H ( M ) i,j ( M ∈ A n , ≤ i, j ≤ n ) , where H ( M ) i,j is the ( i, j )th entry of the (ordinary) height matrix of M . SeeFigure 10 for an example. The following theorem is an equivalent version ofTheorem 11 formulated for these matrices. D. ROMIK Fig. 10. The symmetrized height matrix of the ASM from Figure 7. Theorem 11 ′ . Let G ( x, y ) = x + y − F ( x, y ) , where F is defined inTheorem 11. For each n let M n be a P n Dom -random ASM of order n , and let H ∗ n = H Sym ( M n ) = ( h ∗ i,j n ) ni,j =0 be its associated symmetrized height matrix.Then as n → ∞ we have the convergence in probability max ≤ i,j ≤ n (cid:12)(cid:12)(cid:12)(cid:12) h ∗ i,jn n − G ( i/n, j/n ) (cid:12)(cid:12)(cid:12)(cid:12) P −→ n →∞ . We remark that it would have been possible to work with symmetrizedheight matrices right from the beginning. In that case the large deviationanalysis would have lead directly to Variational Problem 2 without goingfirst through Variational Problem 1. [Note that the limiting symmetrizedheight function G ( x, y ) can also be written as G ( x, y ) = y − g ∗ y ( x ), where g ∗ y is the solution to Variational Problem 2.] 6. Back to domino tilings. We now recall some basic facts from [6] aboutdomino tilings of the Aztec diamond AD n , their height functions, and theirconnection to alternating sign matrices and their height matrices. This willenable us to use our previous results to reprove the Cohn–Elkies–Propplimit shape result for the height function of a uniformly random dominotiling of AD n as n → ∞ .Let G = G (AD n ) be the directed graph whose vertex set is V (AD n ) = { ( i, j ) ∈ Z : | i | + | j | ≤ n + 1 } , and where the adjacency relations are( i , j ) → ( i , j ) ⇐⇒ j = j and i − i = ( − n + i + j , or i = i and j − j = ( − n + i + j +1 . We call G (AD n ) the Aztec diamond graph . Note that its adjacency structureis the standard nearest-neighbor graph structure induced from Z , where inaddition edges are directed according to a checkerboard parity rule, namely, RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX (a) (b) Fig. 11. (a) The Aztec diamond graph of order 3; (b) the normalized height functioncorresponding to the tiling from Figure 1. that if a checkerboard coloring is imposed on the squares [ n, n + 1] × [ m, m +1] in the lattice dual to Z , then the nearest-neighbor edges u → v are alldirected such that a traveller crossing the directed edge will see a blacksquare on her left; see Figure 11(a).Define a height function to be any function η on V (AD n ) such that forany edge u → v in G (AD n ) we have η ( u ) − η ( v ) = 1 or − , and such that η ( u ) − η ( v ) = 1 whenever u → v is one of the boundary edges.A height function η on V (AD n ) is called normalized if η ( − n, 0) = 0.It is known that any domino tiling T of AD n determines a unique nor-malized height function η T by the requirement that for any directed edge u → v we have η T ( u ) − η T ( v ) = (cid:26) − , the segment ( u, v ) crosses a domino tile in T ,1 , otherwise.Conversely, any normalized height function η is of the form η T for somedomino tiling. See Figure 11(b).Another important fact concerns the beautiful connection, discovered byElkies et al. [6], between height functions of domino tilings of AD n andheight matrices of ASMs: each normalized height function η on V (AD n )is essentially comprised of the superposition of two (symmetrized) heightmatrices H Sym ( A ) , H Sym ( B ) where A is an ASM of order n and B is anASM of order n + 1. More precisely, H Sym ( A ) and H Sym ( B ) can be recovered D. ROMIK from η by H Sym ( A ) i,j = η ( − n + 1 + i + j, − i + j ) − , (26) H Sym ( B ) i,j = η ( − n + i + j, − i + j )2(27)(note the slight difference from the formulas in [6] due to a difference in thecenter of the coordinate system used). This correspondence defines a one-to-one mapping from the set of domino tilings of AD n to the set of pairs ( A, B )where A ∈ A n and B ∈ A n +1 . The pairs ( A, B ) which are obtained via thismapping are exactly the so-called compatible pairs defined by Robbins andRumsey [29]: A and B are called compatible if the (nonsymmetrized) heightmatrices H ( A ) , H ( B ) satisfy the conditions H ( B ) i,j ≤ H ( A ) i,j ,H ( B ) i +1 ,j +1 − ≤ H ( A ) i,j ,H ( A ) i,j ≤ H ( B ) i +1 ,j ,H ( A ) i,j ≤ H ( B ) i,j +1 . It was also shown in [29] that for a given ASM A ∈ A n , the number of B ∈ A n +1 that are compatible with A is equal to 2 N + ( A ) . Combined withthe formula for the number of domino tilings of AD n , this implies that if T is a uniformly random domino tiling of AD n , and ( A, B ) is the associatedpair of compatible ASMs, then the random ASM A is distributed accordingto the domino measure P n Dom (of course, this provides the explanation forour choice of name for this measure).We now combine Theorem 11 ′ with the above discussion to easily obtainthe following result, originally proved in [2]. Theorem 12. For each n ≥ , let T n be a uniformly random dominotiling of AD n , and let η n = η T n be its associated height function. Then as n → ∞ we have the convergence in probability max ( i,j ) ∈ V ( AD n ) (cid:12)(cid:12)(cid:12)(cid:12) n η n ( i, j ) − R ( i/n, j/n ) (cid:12)(cid:12)(cid:12)(cid:12) P −→ n →∞ , where R ( u, v ) = 2 G (cid:18) u − v + 12 , u + v + 12 (cid:19) ( | u | + | v | ≤ , and G is defined in Theorem 11 ′ . Proof. For pairs ( i, j ) ∈ V (AD n ) for which i + j + n is odd, the prox-imity of n − η n ( i, j ) to R ( i/n, j/n ) follows from (26). For other pairs ( i, j ),apply the previous observation to any pair ( i ′ , j ′ ) adjacent to ( i, j ) and usethe facts that | η n ( i, j ) − η ( i ′ , j ′ ) | ≤ R is a continuous function. (cid:3) RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX 7. Concluding remarks. Relation to the arctic circle theorem. Theorem 12 implies a weakform of the arctic circle theorem (Theorem 1): first, since inside the arcticcircle the limit shape function R ( u, v ) is not a linear function, it follows thatthe frozen region cannot extend in the limit into the arctic circle, which is“half” of the theorem. In the other direction, we get only a weaker state-ment that outside the arctic circle we can have in the limit at most o ( n )“nonfrozen” dominoes, since that is what the linearity of the limiting heightfunction in that region implies.It is interesting to contrast this with the square Young tableaux problem.There, too, the large deviation approach gave only a bound in one directionon the behavior of the square Young tableau along the boundary of thesquare. However, Pittel and Romik managed to prove the other directionusing an additional combinatorial argument (inspired by a method of Vershikand Kerov [33]). It would be interesting to see whether one can emulate thisapproach in the present case to get a new proof of the arctic circle theorem.A similar question applies to the problem of random boxed plane partitionsstudied by Cohn, Larsen and Propp [4], where again the limit shape theoremfor the height function does not imply an arctic circle result in its strongform.7.2. Other arctic circles and more general arctic curves. In this paperwe have shown that two so-called arctic circle phenomena, namely thoseappearing in the contexts of random domino tilings of the Aztec diamondand of random square Young tableaux, are closely related, in the sense thatthe limit shape results underlying them can be given a more or less unifiedtreatment using the techniques of large deviation theory and the calculus ofvariations, and that the derivations in both cases result in nearly identicalcomputations and formulas. Note that these are not the only combinatorialmodels in which arctic circles appear. Other examples known to the authorinclude the shape of a uniformly random boxed plane partition derived byCohn, Larsen and Propp [4] and the arctic circle theorem for random groves,due to Petersen and Speyer [25]. One might therefore wish to extend theinsights of the present paper to these other models. The treatment of boxedplane partitions in [4] is already based on a large deviations analysis, and infact the variational problem studied there seems to be quite closely relatedto the variational problems studied here and in [26]. Therefore, it shouldbe relatively straightforward to use the techniques presented here to givea new derivation of the solution to the variational problem from [4] (whichin particular would provide a fully satisfactory answer to Open Question 6.3from that paper). D. ROMIK The analysis of random groves, on the other hand, is based on generatingfunction techniques, and it is not clear how to apply the ideas presentedhere to that setting.It is also worth mentioning that there is a large literature on the subjectof limit shapes of various classes of random combinatorial objects, and tilingmodels in particular, where one encounters in many cases a spatial phasetransition between a “frozen” and a ”temperate” region. The equations gov-erning such limit shapes can in general lead to a much more diverse familyof noncircular “arctic curves” describing the shape of the interface betweenthe frozen and temperate regions. For details, see, for example, the papers[3, 21, 22].7.3. Uniformly random ASMs. One reason why the methods and ideaspresented in this paper may be considered worthy of attention is somewhatspeculative in nature. It pertains to the potential future applicability ofthese methods and ideas to a well-known open problem on alternating signmatrices: that is, the problem of finding the limiting shape of a uniformly random ASM of high order. Here, “limit shape” is usually taken to refer tothe shape of the region in which the nonzero entries cluster (the “temperateregion”), although one could also ask (as we have done here in the case of P n Dom -random ASMs) about the limiting shape of the height matrix, whichalso contains useful information about the behavior of the ASM inside thetemperate region.Important progress on this question was made recently by Colomo andPronko [5], who conjectured the explicit formula x + y + | xy | = | x | + | y | for the limit shape of the boundary of the temperate region in a uniformlyrandom ASM (Figure 12), and provided a heuristic derivation of this con-jectured formula based on certain natural, but still conjectural, analyticassumptions. Fig. 12. The Colomo–Pronko conjectured limit shape for uniformly random alternatingsign matrices. RCTIC CIRCLES, DOMINO TILINGS AND SQUARE YOUNG TABLEAUX In view of this state of affairs, it is worth noting that the ideas presentedin this paper seem to be rather suitable for attacking this challenging openproblem. There is only one main “missing piece” (albeit possibly a verysubstantial one) in our understanding. The idea is to replace Theorem 3,which is the combinatorial observation which lies at the heart of the largedeviations analysis, with an analogous statement that holds for the uniformmeasure on the set A n of ASMs of order n . This statement is given in the fol-lowing theorem, whose proof follows similar lines to the proof of Theorem 3and is omitted. Theorem 13. Let P Unif denote the uniform measure on the set of ASMsof order n . For a positive integer k and integers x < x < · · · < x k , denoteby α k ( x , . . . , x k ) the number of monotone triangles of order k with bottomrow ( x , . . . , x k ) . Then, in the notation of Theorem 3, we have P Unif [ M ∈ A n : ( X k (1) , . . . , X k ( k )) = ( x , . . . , x k )]= 1 |A n | α k ( x , . . . , x k ) α n − k ( y , . . . , y n − k ) . Unfortunately, while a formula for |A n | is known (see [1]), the function α k seems much more difficult to understand (and in particular, to derive asymp-totics for) than the Vandermonde function ∆, and this is the piece that ismissing when one tries to duplicate our analysis to the setting of uniformlyrandom ASMs. Nevertheless, the function α k has recently been the subjectof several very fruitful studies. Fischer [9] derived the following beautiful“operator formula” for α k : α k ( x , . . . , x k ) = (cid:20) Y ≤ i Bressoud, D. M. (1999). Proofs and Confirmations: The Story of the AlternatingSign Matrix Conjecture . Mathematical Association of America, Washington, DC.MR1718370[2] Cohn, H. , Elkies, N. and Propp, J. (1996). Local statistics for random dominotilings of the Aztec diamond. Duke Math. J. Cohn, H. , Kenyon, R. and Propp, J. (2001). A variational principle for dominotilings. J. Amer. Math. Soc. Cohn, H. , Larsen, M. and Propp, J. (1998). The shape of a typical boxed planepartition. New York J. Math. Colomo, F. and Pronko, A. G. (2010). The limit shape of large alternating signmatrices. SIAM J. Disc. Math. Elkies, N. , Kuperberg, G. , Larsen, M. and Propp, J. (1992). Alternating signmatrices and domino tilings. J. Algebraic Combin. Estrada, R. and Kanwal, R. P. (2000). Singular Integral Equations . Birkh¨auser,Boston, MA. MR1728075[8] Finch, S. R. (2003). Mathematical Constants . Encyclopedia of Mathematics and ItsApplications . Cambridge Univ. Press, Cambridge. MR2003519[9] Fischer, I. (2006). The number of monotone triangles with prescribed bottom row. Adv. in Appl. Math. Fischer, I. (2007). A new proof of the refined alternating sign matrix theorem. J. Combin. Theory Ser. A Fischer, I. (2010). Linear relations of refined enumerations of alternating sign ma-trices. Unpublished manuscript. Available at arXiv:1008.0527v1.[12] Fischer, I. (2010). The operator formula for monotone triangles—simplified proofand three generalizations. J. Combin. Theory Ser. A Fischer, I. (2011). Refined enumerations of alternating sign matrices: Monotone( d, m )-trapezoids with prescribed top and bottom rows. J. Algebraic Combin. Fischer, I. and Romik, D. (2009). More refined enumerations of alternating signmatrices. Adv. Math. Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Prod-ucts , 6th ed. Academic Press, San Diego, CA. MR1773820[16] Jockusch, W. , Propp, J. and Shor, P. (1995). Random domino tilings and the arc-tic circle theorem. Unpublished manuscript. Available at arXiv:math/9801068.[17] Johansson, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherelmeasure. Ann. of Math. (2) Johansson, K. (2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields Johansson, K. (2005). The arctic circle boundary and the Airy process. Ann. Probab. Kasteleyn, P. W. (1961). The statistics of dimers on a lattice. I. The number ofdimer arrangements on a quadratic lattice. Physica Kenyon, R. and Okounkov, A. (2007). Limit shapes and the complex Burgersequation. Acta Math. [22] Kenyon, R. , Okounkov, A. and Sheffield, S. (2006). Dimers and amoebae. Ann.of Math. (2) Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Youngtableaux. Adv. Math. Mills, W. H. , Robbins, D. P. and Rumsey, H. Jr. (1983). Alternating sign ma-trices and descending plane partitions. J. Combin. Theory Ser. A Petersen, T. K. and Speyer, D. (2005). An arctic circle theorem for Groves. J. Combin. Theory Ser. A Pittel, B. and Romik, D. (2007). Limit shapes for random square Young tableaux. Adv. in Appl. Math. Porter, D. and Stirling, D. S. G. (1990). Integral Equations . Cambridge Univ.Press, Cambridge. MR1111247[28] Propp, J. (2001). The many faces of alternating-sign matrices. In Discrete Mod-els: Combinatorics, Computation, and Geometry (Paris, 2001) . Discrete Math.Theor. Comput. Sci. Proc. AA Robbins, D. P. and Rumsey, H. Jr. (1986). Determinants and alternating signmatrices. Adv. Math. Temperley, H. N. V. and Fisher, M. E. (1961). Dimer problem in statisticalmechanics—an exact result. Philos. Mag. (8) Valk´o, B. (2006). Private communication.[32] Vershik, A. M. and Kerov, S. V. (1977). Asymptotics of the Plancherel measureof the symmetric group and the limiting shape of Young tableaux. Soviet Math.Dokl. Vershik, A. M. and Kerov, S. V. (1985). Asymptotic of the largest and typicaldimensions of irreducible representations of the symmetric group.