Limit shape of minimal difference partitions and fractional statistics

The class of minimal difference partitions MDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q\ge 0. In a recent series of papers by A. Comtet and collaborators, the MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classic Bose-Einstein (q=0) and Fermi-Dirac (q=1) cases. This was done by formally allowing values q \in (0,1) using an analytic continuation of the limit shape of the corresponding Young diagrams calculated for integer q. To justify this"replica-trick", we introduce a more general model based on a variable MDP-type condition encoded by an integer sequence (q_i), whereby the (limiting) gap q is naturally interpreted as the Cesaro mean of (q_i). In this model, we find the family of limit shapes parameterized by q \in [0,\infty) confirming the earlier answer, and also obtain the asymptotics of the number of parts.

A partition λ = (λ 1 , λ 2 , . . . ) ∈ Λ is succinctly visualized by its Young diagram Υ λ formed by left-and bottom-aligned column blocks with λ 1 , λ 2 , . . . unit square cells, respectively. In particular, the area of the Young diagram Υ λ equals the partition weight N(λ). The upper boundary of Υ λ is a non-increasing step function Y λ : [0, ∞) → Z + := {0, 1, 2, . . . } (see Fig. 1 for illustration). Note that inf{t ≥ 0 : Y λ (t) = 0} coincides with the length K(λ). Theory of integer partitions is a classical branch of discrete mathematics and combinatorics dating back to Euler, with further fundamental contributions due to Hardy, Ramanujan, Rademacher and many more (see [3] for a general background). The study of asymptotic properties of random integer partitions (under the uniform distribution) was pioneered by Erdős & Lehner [12], followed by a host of research which in particular discovered a remarkable result that, under a suitable rescaling, the Young diagrams of typical partitions of a large integer n are close to a certain deterministic limit shape. For strict partitions (i.e., with distinct parts) this result was (implicitly) contained already in [12]; for partitions without any restrictions, the limit shape was first identified by Temperley [33] in relation to the equilibrium shape of a growing crystal, and obtained more rigorously much later by Vershik (as pointed out at the end of [37]) using some asymptotic estimates by Szalay & Turán [32]. An alternative proof in its modern form was outlined by Vershik [34] and elaborated by Pittel [28], both using the conditioning device 1 based on a suitable randomization of the integer n being partitioned.
To date, many limit shape results are known for partitions subject to various restrictions, see a review in the recent paper by DeSalvo & Pak [11]. Deep connections between statistical properties of quantum systems (where discrete random structures naturally arise due to quantization) and asymptotic theory of random integer partitions are discussed in a series of papers by Vershik [34,35]. Note that the idea of conditioning in problems of quantum statistical mechanics was earlier promoted by Khinchin [21] who advocated systematic use of local limit theorems of probability theory as a tool to prove the equivalence of various statistical ensembles in the thermodynamic limit.
From the point of view of statistical mechanics, it is conventional 2 to interpret the integer partition λ = (λ i ) ∈ Λ as the energy spectrum in a sample configuration (state) of quantum gas, with K(λ) = #(λ i > 0) particles and the total energy i λ i = N(λ). Note that decomposition into a sum of integers is due to the quantization of energy in quantum mechanics, while using unordered partitions corresponds to the fact that quantum particles are indistinguishable. In this context, the limit shape of Young diagrams associated with random partitions (for instance, under the uniform measure) is of physical interest as it describes the asymptotic distribution of particles in such ensembles over the energy domain.

Minimal difference partitions
For a given q ∈ N 0 , the class of minimal difference partitions with gap q, denoted MDP(q), is the set of integer partitions λ = (λ 1 , λ 2 , . . . ) subject to the restriction λ i − λ i+1 ≥ q whenever λ i > 0. Two important special cases of the MDP(q) are furnished by the values q = 0 corresponding to plain partitions (i.e., with no restrictions), and q = 1 leading to strict partitions (i.e., with different parts).
In this paper, we propose a natural generalization of the MDP property as follows.
Definition 1.1. For a given sequence q = (q i ) i∈N 0 of non-negative integers (with the convention that q 0 ≥ 1), we define Λ q ≡ MDP(q) to be the set of all integer partitions λ = (λ i ) subject to the variable MDP-type condition where k is the number of (non-zero) parts in the partition λ. By convention, the empty partition ∅ satisfies (1.1).
Remark 1.1. For i = k, the inequality (1.1) specializes to λ k − λ k+1 ≡ λ k ≥ q 0 . That is to say, the smallest part of the partition λ = (λ i ) is required to be not less than q 0 ≥ 1 (which really poses a restriction only if q 0 > 1).
Remark 1.2. Alternatively, one could consider partitions subject to similar restrictions as (1.1) but in the reverse order relative to the sequence q, However, the model (1.1) is preferable in view of the physical interpretation of parts λ i as successive energy levels in a configuration (state) of a quantum system [35], which makes it more natural to enumerate the energy gaps starting from the minimal level λ k = min{λ i : λ i > 0}. One more motivation for choosing the model (1.1) is its earlier appearance (without any name) in work [4] devoted to partition bijections.
Throughout the paper, we impose the following Assumption 1.1. The sequence q = (q i ) i∈N 0 (q i ∈ N 0 , q 0 ≥ 1) satisfies the asymptotic regularity condition with some q ≥ 0 and 0 ≤ β < 1.
Note that under Assumption 1.1 the sequence q = (q i ) has a well-defined Cesàro mean, referred to as the limiting gap, In the case q = 0, the asymptotic relation (1.2) accommodates sequences (Q k ) that are irregularly growing (provided the growth is sublinear) or even bounded (β = 0), including the case Q k ≡ 1 corresponding to plain (unrestricted) integer partitions.
For q = 0 (when the leading term in (1.2) vanishes), it is still possible to derive the limit shape results under our standard Assumption 1.1. However, to obtain the asymptotics of the typical MDP length K(λ), more regularity should be assumed by specifying the behaviour of the remainder term O(k β ).
Remark 1.4. The utterly degenerate caseq = 0 andβ = 0 in Assumption 1.2 is equivalent to Assumption 1.1 with q = 0 and β = 0. In this case, we have Q k = O(1) as k → ∞, and since q i ∈ N 0 , this implies that q i = 0 for all sufficiently large i. Clearly, the first few non-zero terms in the sequence q = (q i ) (i.e., in the MDP conditions (1.1)) do not affect any limiting results, and so effectively such a model is identical with the classical case of unrestricted partitions (q 0 = 1 and q i ≡ 0 for i ∈ N).

Main result
For n ∈ N 0 , consider the subset Λ q (n) = Λ q ∩Λ(n) comprising MDP(q) partitions of weight N(λ) = n. For example, the partition λ = (8, 6, 6, 5, 4, 2, 2, 1, 1, 0, 0, . . . ) used in Fig. 1 fits into the MDP-space Λ q (35) with the alternating sequence q = (1, 0, 1, 0, 1, 0, . . . ). Suppose that each (non-empty) space Λ q (n) is endowed with uniform probability measure denoted ν q n . We are interested in asymptotic properties (as n → ∞) of this and similar measures on MDP spaces; in particular, we find the limit shape of properly scaled Young diagrams associated with partitions λ ∈ Λ q (n) and prove exponential bounds for deviations from the limit shape. Let us state one of our main results, slightly simplifying the notation as compared to the more general case treated in Section 4. For every q ≥ 0, define the function and let T q := inf{t > 0 : ϕ(t; q) = 0}; that is, T q is the unique root of the equation (with the convention T 0 := +∞). The area under the graph of ϕ(t; q) is computed as where Li 2 (·) denotes the dilogarithm function (see, e.g., [24, p. 1]), Note that Li 2 (1) = ζ(2) = π 2 /6. It is easy to check from (1.6) that lim q↓0 qT 2 q = 0, so using (1.7) we obtain Finally, observe that, setting x = e −Tq in the well-known identity 3 [24, Eq. (1.11), p. 5] and using the equation (1.6), the expression (1.7) is rewritten in a more appealing form, where the terms on the right-hand side can be given a meaningful geometric interpretation (see details in Section 4.4).
Remark 1.5. It is common to scale Young diagrams via reducing their area n to 1 [35]. In our case, this leads to the additional rescaling in the expression of the limit shape (see (1.12)). Instead, it is more natural to work with the intrinsic scaling (1.13) to produce a simpler equation for the limit shape (1.14) but where the limiting area ϑ 2 q varies with q (see (1.11)). See the precise corresponding assertions in Section 4. Example 1.1. Let us specialize the notation introduced before Theorem 1.1 for a few simple values of q ≥ 0, including all cases where closed expressions for T q and ϑ q in elementary functions are available.

MDP and fractional statistics
The special case of the MDP(q) model with a constant sequence q i ≡ q ∈ N 0 in (1.1) was considered in a series of papers by Comtet et al. [7,8,9] in connection with fractional exclusion statistics of quantum particle systems (see [20], [23] or [25] for a "physical" introduction to this area). These authors obtained the limit shape of MDP(q) using a physical argumentation. In particular, it was observed that the analytic continuation of the limit shape, as a function of q ∈ N 0 , into the range q ∈ (0, 1) (the so-called replica trick) may be interpreted as a quantum gas obeying fractional exclusion statistics, thus furnishing a family of probability measures "interpolating" between the Bose-Einstein statistics (q = 0) and the Fermi-Dirac statistics (q = 1).
In the present work, we provide a combinatorial justification of this physical construction by working with a more general MDP(q) model satisfying Assumption 1.1. In addition to many deterministic examples with such a property, the assumption (1.2) (and hence (1.3)) holds almost surely for sequences of independent random variables q = (q i ) satisfying mild conditions, thus providing a stochastic version of the MDP(q) model (see Section 6 below).
As was observed by Comtet et al. [7], another model of statistical physics leading to the MDP-type constraint is the one-dimensional quantum Calogero model with harmonic confinement (see [29] for a review and further references therein), defined by the Hamiltonian of a k-particle system with spatial positions (x i ) k i=1 on a line, This model is exactly solvable, and the solution can be expressed in terms of the pseudoexcitation numbers λ i satisfying the condition λ i − λ i+1 ≥ q, with a positive real q.
As is common in such models (cf. [18]), an analogue of Pauli's exclusion principle is not strictly local for models MDP(q) with sequences q = (q i ) not degenerating to the trivial sequences q i ≡ 0 or q i ≡ 1 (i ∈ N). Indeed, the occurrence of part λ i = j rules out a few adjacent values, that is, but the actual index k − i is determined by the entire partition λ = (λ i ) through the rank of the part λ i = j among all (ordered) parts λ i , together with the total number k of non-zero parts in λ. Remark 1.6. Heuristically, the requirement λ i − λ i+1 ≥ q with q ∈ (0, 1] may be interpreted, at least for integer m := q −1 , as saying that λ i − λ i+m ≥ 1 as long as λ i > 0, that is, to prohibit more than m = q −1 equal parts; in other words, no part counts bigger than q −1 are allowed. For q = 1 this indeed translates as only strict partitions being permissible. In the general case, this interpretation turns out to be true for the expected part counts (see [20, § 5.2]); however, literal restriction that the part counts do not exceed q −1 leads to a different model called Gentile's statistics [20, § 5.5]. The limit shape of partitions under Gentile's statistics was found in [26, § 9] (see also [39] where a rigorous proof is given).
The rest of the paper is organized as follows. In Section 2, several measures on minimal difference partitions are introduced, and certain relations between them are stated. Section 3 is devoted to finding the typical length of MDPs. In Section 4 the main results concerning the limit shape of MDPs, both with a restricted and unrestricted length growth, are proved. In fact, we obtain sharp exponential bounds for deviations from the limit shape. Section 5 describes an alternative approach to the limit shape based on a partition bijection that effectively removes the MDP-constraint. In Section 6, we extend our results to the case of random sequences q. Finally, the Appendix contains proof of the two technical propositions stated in Section 2, which establish the equivalence of ensembles.
2. Probability measures on the MDP spaces 2.1. Basic definitions and notation In this paper, we shall use several probability measures on MDPs and other partition spaces. In the present section we describe them and establish some properties. First we introduce notation for some functionals on partitions we shall need. If one fixes a probability measure on partitions, these functionals become random variables.
Let λ = (λ 1 , λ 2 , . . . ) (λ 1 ≥ λ 2 ≥ · · · ≥ 0) be an integer partition, λ ∈ Λ. Recall that N(λ) := λ 1 + λ 2 + · · · is referred to as the weight of λ by and its length (number of parts) by K(λ) := #{λ i ∈ λ : λ i > 0}. An equivalent description of a partition λ can be given in terms of the consecutive differences D j (λ) = λ j − λ j+1 ; obviously, Evidently, the map t → Y λ (t) is non-increasing, piecewise constant, and right-continuous. From (2.1), it is also easy to see that Y λ (t) = λ ⌊t⌋+1 (t ≥ 0), with ⌊·⌋ denoting the floor function (i.e., integer part). The Young diagram Υ λ of a partition λ is defined as the closure of the planar set That is to say, the Young diagram Υ λ is the union of (left-and bottom-aligned) column blocks with λ 1 , λ 2 , . . . unit squares, respectively; in particular, the function t → Y λ (t) defines its upper boundary (cf. Section 1.1). We shall often identify the Young diagram Υ λ with the (graph of the) function Y λ (t) (see Fig. 1). The measure most important for us is the aforementioned uniform measure ν q n on the set Λ q (n): The space Λ q (n) can be further decomposed as a disjoint union of the sets Λ q (n, k) := {λ ∈ Λ q (n) : K(λ) = k}, and one can introduce the uniform measures on these spaces, Note that ν q n,k can be viewed as the measure ν q n conditioned on the event {K(λ) = k}; indeed, for any λ ∈ Λ q (n, k), This conditional measure is somewhat simpler than ν q n itself, since there exists a product expression for the Laplace generating function of p q (n, k) with respect to n (for any fixed k).
To establish such an expression, the following simple observation is useful. Define Then the MDP(q) condition (1. one-to-one correspondence with the set D q (k). Moreover, using the second of the formulas , where we set 2)); moreover, the asymptotic condition (1.2) implies that, for q ≥ 0, 3) is similar to that of multiplicative measures introduced by Vershik [34]. However, there are some distinctions from multiplicative measures. Firstly, the partition length K(λ) must be fixed to obtain independence. Secondly, the role of the part counts which become independent after randomization of N(λ) = n is played here by the differences D j (λ).
Let us define an auxiliary probability measure µ q Note that, for every z > 0, the measure µ q z,k conditioned on the event {N(λ) = n} coincides with the uniform measure ν q n,k on the space Λ q (n, k); indeed, according to (2.6) we have, for any λ ∈ Λ q (n, k), The following fact will be instrumental below.
In particular, the expected values are given by Proof. The claim easily follows from the representation of N(λ) through (D j (λ)) (see (2.1)) and the product structure of the Laplace generating function (2.3).
Similarly, we can assign the weight e −zN (λ) to each partition . (2.10) Note that the series (2.10) converges for all z > 0, since it is bounded by the convergent series This way, we get the probability measure Similarly to (2.7), it is easy to check that the measure µ q z conditioned on {N(λ) = n} coincides with the uniform measure ν q n on Λ q (n), Furthermore, the definition (2.11) implies We finish this subsection by a comment linking the above MDP spaces and probability measures on them with the general nomenclature of ensembles in statistical mechanics (see, e.g., the monographs by Huang [19] or Greiner et al. [17]). Under the quantum interpretation of integer partitions λ = (λ i ) ∈ Λ briefly mentioned in Section 1.1, the MDP(q) restriction determines the exclusion rules for permissible energy levels (λ i ). In general, the weight N(λ) (total energy) and length K(λ) (number of particles) are random. Fixing one or both of these parameters leads to different measures on the corresponding spaces, and therefore determines different ensembles. In particular, a completely isolated system, with fixed N(λ) = n and K(λ) = k and under uniform measure ν q n,k on the corresponding Micro-canonical: ν q n,k on Λ q (n, k) Fig. 2: Schematic diagram illustrating the relation between different MDP-ensembles. The integer parameters n and k are interpreted as the total energy of the (quantum) system and the number of particles, respectively. The arrows "heat bath" and "particle bath" indicate that fixation of energy or the number of particles is lifted.
space Λ q (n, k), has the meaning of micro-canonical MDP ensemble. When, say, the fixation N(λ) = n is lifted (which may be thought of as connecting the system to a heat bath, whereby thermal equilibrium is settled through exchange of energy with the bath), we get an enlarged space Λ( · , k) with the measure µ q z,k , which is interpreted as the canonical ensemble, with a fixed number of particles k. Furthermore, removing the latter constraint (which, similarly, is achieved by putting the canonical ensemble into a particle bath allowing free exchange of particles) leads to the space Λ q with the measure µ q z , which is referred to as the grand canonical ensemble (see the schematic diagram in Fig. 2).
Note however that the space Λ q (n) (i.e., with a fixed energy N(λ) = n and endowed with uniform measure ν q n ), which is most natural from the combinatorial point of view, is missing in this picture; indeed, it may not be physically meaningful to talk about systems with fixed energy and free number of particles. But logically, it is perfectly possible to interchange the order of relaxations described above and first lift the condition K(λ) = k by connecting the micro-canonical system to a particle bath; we take the liberty to call the resulting ensemble meso-canonical, 4 indicating an intermediately coarse partitioning of the phase space (cf. [13]). Finally, removing the remaining constraint N(λ) = n (by connecting the system further to a heat bath) we again obtain the grand canonical ensemble.

Asymptotic equivalence of ensembles
For q ≥ 0, define the function (2.14) Note that the value ϑ q (T q ) coincides with the notation ϑ q introduced in (1.11).
The following curious identity will be explained in Section 4.4.
The next proposition establishes an asymptotic link between the measures µ q z and ν q n . Proposition 2.3. Suppose that the sequence q satisfies the condition (1.2). Let {A z } z>0 be a family of subsets of the space Λ q such that, for some positive constant κ, Then there exists a sequence (z n ) such that There is a similar connection between the measures µ q z,k and ν q n,k , provided that z ↓ 0, k → ∞ and n → ∞ in a coordinated manner.
These two propositions are instrumental for our method; their proof, being rather technical, is postponed until Appendix A.

Number of parts in a typical MDP
In this section, our ultimate goal is to show that if Assumption 1.1 holds then, under the measures ν q n on the MDP-space Λ q (n), the typical length K(λ) (i.e., the number of parts) of a partition λ ∈ Λ q (n) of large weight N(λ) = n is concentrated around c √ n (with a suitable constant c > 0) if q > 0, or grows slightly faster than √ n if q = 0. To this end, we will first study the distribution of K(λ) under the measure µ q z in the space Λ q .

Preparatory lemmas
For z > 0, denote where Q k is given by (1.2). For every z > 0, the sequence (η k (z)) k≥1 is decreasing, and in particular Thus, the set {k : η k (z) ≥ 1} is always finite (possibly empty). Define Remark 3.1. Note that lim z↓0 η k (z) = +∞ for any fixed k ∈ N, and so k * (z) > 1 for all z > 0 small enough.
First, let us record a few auxiliary statements that do not depend on Assumption 1.1.
(a) For every z > 0, we have
(b) For k ∈ N, let z = ζ k be the (unique) solution of the equation From the formulas (3.1) and (3.6), it is clear that the sequence (ζ k ) k≥1 is decreasing and, moreover, are the two maxima of the sequence µ q z {K(λ) = j} j≥1 , whereas for z ∈ (ζ k+1 , ζ k ) the unique maximum of this sequence is attained exactly at j = k. Hence, k * (z) ≡ k for z ∈ (ζ k+1 , ζ k ], that is, z → k * (z) is a non-increasing (left-continuous) step function with unit downward jumps at points ζ k (k ≥ 2). Since lim k→∞ ζ k = 0, it also follows that lim z↓0 k * (z) = +∞.
where T q is defined in (2.14).
Comparing this with equation (2.14), observe that kζ k = Tq k , whereq k := k −1 Q k → q > 0 as k → ∞, due to the limit (1.3), and therefore lim k→∞ Tq k = T q , thanks to continuity of the mapping q → T q mentioned after the definition (2.14). To see why this implies (3.12), recall from the proof of Lemma 3.1(b) that k * (z) ≡ k for z ∈ (ζ k+1 , ζ k ] (k ∈ N) and the limit z ↓ 0 is equivalent to k → ∞. Hence, Furthermore, by a standard perturbation analysis it is easy to estimate the corresponding remainder term in the limit (3.12). Indeed, setting δ k := kζ k − T q → 0 and using the asymptotic relation (1.2), we can rewrite (3.15) in the form which yields, in view of the identity (2.14), that δ k = O(k β−1 ). In turn, for ζ k+1 < z ≤ ζ k we get which implies the last inequality in (3.13), since ε > 0 can be taken arbitrarily close to 0.
On the other hand, from (2.14) we also have η k * +1 (z) < 1, that is, Furthermore, using the asymptotic bound (1.2) for k = k * (with q = 0) and the estimate (3.16), we obtain Substituting this into (3.17), it is easy to see that and the first inequality in (3.13) is proved.
3.3. Asymptotics of K(λ) in the space Λ q : case q = 0 When Assumption 1.1 holds with q = 0, the asymptotics for k * (z) as z ↓ 0 cannot be obtained, as was mentioned in Remark 3.2. So there is no hope to find exponential bounds for K(λ) to fit into an interval of order smaller than z −1 , as in (3.18). Nevertheless we can still find an interval such that K(λ) does not hit it with an exponentially small µ q z -probability, as z ↓ 0. To this end, we need some additional notation.
Fix γ ∈ (0, 1) and define the function Recalling that s k ≥ k (see after formula (2.4)), from the definition (3.27) it follows that On the other hand, it is clear that k γ (z) → ∞ as z ↓ 0. Actually we can tell more.
In the case q = 0, under the refined Assumption 1.2 withq > 0 (see (1.4)) one can prove the following analogue of the exponential bound (3.18): for any c > 0 and γ ∈ 0, 1 2 β , Here k * = k * (z) is again defined by (3.2) but now has the refined asymptotics (cf. (3.13)) The exponential bound (3.40) together with the asymptotic formula (3.41) immediately imply the law of large numbers for the number of parts (cf. Corollary 3.5): for any ε > 0, Formally, these results do not cover the utterly degenerate caseq = 0,β = 0 in the asymptotic formula (1.4) of Assumption 1.2; however, as explained in Remark 1.4, it is equivalent to the classical case of unrestricted partitions, where the asymptotic behaviour of K(λ) (under the measure µ z on Λ) is described by the limit theorem [15] lim z↓0 The asymmetry of the limiting distribution (3.42) (i.e., exponential tail on the right and super-exponential tail on the left) explains the appearance of the two claims in Theorem 3.7.
(a) If q > 0 then there exists a sequence (k n ) satisfying the asymptotic relation such that, for any a > 0, lim sup n→∞ n γ−1/2 log ν q n λ ∈ Λ q (n) : |K(λ) − k n | > a n (1−γ)/2 < 0.  (b) Consider the set A z = {K(λ) < z −1 log log 1 z or K(λ) > k γ (z)}. By Theorem 3.7, the set A z satisfies the condition (2.16) of Proposition 2.3. Moreover, if the asymptotic relation (2.17) with q = 0 holds for a sequence z n , then the set referred to in (3.45) is a subset of A zn , at least for n large enough, because Thus, the required relation (3.45) follows from (2.18).
Similarly as before, Theorem 3.8 with q > 0 implies the law of large numbers for K(λ) under the measure ν q n , analogous to Corollary 3.5. Corollary 3.9. Let Assumption 1.1 hold with q > 0. Then, for any ε > 0, lim n→∞ ν q n λ ∈ Λ q (n) : If q = 0 then, under Assumption 1.2 withq > 0 (see (1.4)), one can deduce in a similar fashion the law of large numbers for K(λ): for any ε > 0, lim n→∞ ν q n λ ∈ Λ q (n) : In fact, an exponential bound for large deviations of K(λ) can be obtained by combining (3.40) with Theorem 3.8(b), but we omit technical details. Finally, ifq = 0 andβ = 0 in (1.4), then the classical limit theorem (under the uniform measure ν n on Λ(n)) states that [12,15] lim n→∞ ν n λ ∈ Λ(n) : Of course, this result implies the law of large numbers, lim n→∞ ν n λ ∈ Λ(n) : which can be formally considered as the limiting case of (3.46) as β ↓ 0. Remark 3.3. To be more precise, the results by Erdős & Lehner [12] and Fristedt [15], quoted above as formulas (3.42) and (3.47), are technically about the maximal part λ 1 , but due to the invariance of the measures µ z and ν n under conjugation of Young diagrams (whereby columns become rows and vice versa; see also Section 5), the random variable λ 1 has the same distribution as the number of parts K(λ).

The parametric family of limit shapes
Mutual independence of the random variables (D j (λ)) k j=1 with respect to the measure µ q z,k (see Lemma 2.1) provides an easy way to find the limit shape for MDPs as z ↓ 0. It is natural to allow the maximal part k to grow to infinity as z approaches 0, where the correct growth rate, as suggested by Theorem 3.4, is of order z −1 when q > 0 and possibly faster, by a logarithmic factor, when q = 0. It turns out that if the condition (1.2) holds and lim z↓0 zk = T < ∞ then µ q z,k -typical partitions λ ∈ Λ q ( · , k) concentrate around the limit shape determined by the function If q = 0 then the expression (4.1) is reduced to which coincides, as one could expect, with the limit shape of plain (unrestricted) partitions subject to the condition zk → T (see [38]). If q = 0, one can also allow zk to grow slowly to infinity as z ↓ 0 (which is actually a typical behaviour), whereby the limit shape is given by the formula (which is formally consistent with (4.2) if we set T = ∞).
Another simplification of formula (4.1) worth mentioning occurs for q > 0 and T = T q (see (2.14)), which determines the typical behaviour of the number of parts in this case (see Theorem 3.4 and the asymptotic formula (3.14)); here, the limit shape (4.1) is reduced to which was already mentioned in Section 1.3 (see (1.14)).

4.2.
The limit shape in the spaces Λ q ( · , k) and Λ q (n, k n ) The exact statement is as follows. Recall that the notation k γ (z) is defined in (3.27).
To obtain the exponential bound (4.5), we use a standard technique often applied in similar problems (see, e.g., [10]). Suppose that zk → T ∈ (0, ∞], and fix t ∈ (0, T ) and ε > 0. In what follows, we always assume that z is small enough so that zk > t and Then for any u ∈ (0, t) where the first inequality is a consequence of assumption (4.10), the second is the exponential Markov inequality, and the last line follows from the additive structure of Y λ (t) and independence of (D j ) k j=1 . Suppose that, for some w ∈ (0, 1) that will be specified later, where we put for short t ∈ (0, ∞). Then for j ≥ t/z we have Applying the elementary inequalities with x := (e u − 1) h(zj) and v := v(w) (see (4.12)), we obtain Hence, for u ≤ min{v(w), t} ≤ jz (4.14) Substituting (4.14) into (4.11) and recalling (4.9), we obtain Since y(w) → 1 as w ↓ 0, we can choose w small enough to make the right-hand side of (4.15) negative. This yields the desired bound for the probability of positive deviations in (4.5). The probability of negative deviations is estimated in the same fashion.
We are now in a position to state and prove our first main result.
Remark 4.1. The assumption τ 2 < 2/q in Theorem 4.2 arises naturally, because if λ ∈ Λ q (n, k) then, due to the MDP condition (1.1), we must have n ≥ s k = 1 2 qk 2 + O(k 1+β ), which yields τ 2 ≤ 2/q. The boundary case τ 2 = 2/q can in principle be realized, but both the formulation and analysis should be more accurate, so we do not consider it with the exception of the important special case q = 0 when additional difficulties can be treated without much effort.
Finally, it is easy to see that the sequence (z n ) of Proposition 2.4 and the sequence (z n ) defined by (4.18) are asymptotically equivalent so can be interchanged in (4.16).

The limit shape in the spaces Λ q and Λ q (n)
Recall that the function ϕ Tq (t; q) is given by (4.3), where T q is defined as the unique solution of the equation (2.14).
Our second main result describes the limit shape under the measure ν q n , that is, without any restriction on the number of parts.
Proof. The claim follows from Theorem 4.3 and Proposition 2.3 by the same argumentation as that used to derive Theorem 4.2 from Theorem 4.1 and Proposition 2.4.

Ground state
Observe that, for q > 0, the area beneath the limit shape t → ϕ Tq (t; q) featured in Theorems 4.3 and 4.4 contains a right-angled triangle ∆ q (shaded in Fig. 4) obtained in the limit from the (rescaled) partitions in Λ q (n) satisfying the hard version of the MDP restrictions (1.1), that is, when all inequalities are replaced by equalities. Thus, we can say that the triangle ∆ q represents the ground state of the MDP(q) system, while the remaining part of the limit shape corresponds to additional degrees of freedom in a ν q n -typical partition. Note that, according to the ν q n -typical asymptotic behaviour of K(λ) described in Corollary 3.9, under the scaling of Theorem 4.4 the horizontal leg of the triangle ∆ q is identified as T q . On the other hand, by the condition (1.3) the slope of the hypotenuse of the triangle is given by q, therefore the vertical leg of ∆ q is found to be qT q ; in particular, the area of ∆ q is 1 2 qT 2 q . Since the total area of the limit shape is ϑ 2 q (see (1.11)), the area of the "free" part is given by This remark helps to clarify the duality identity (2.15) of Lemma 2.2. To this end, consider the triangle∆ q obtained from ∆ q by reflection about the principal coordinate diagonal, that is, with legs qT q (horizontal) and T q (vertical). This triangle may serve as the ground state of a suitable MDP(q) ensemble. The slope of the hypotenuse of∆ q is 1/q, which therefore gives the limiting gap of the space MDP(q). But according to the previous considerations, the legs of the triangle∆ q must have the lengths T 1/q (horizontal) and (1/q) T 1/q (vertical). Comparing these values, we arrive at the identity (2.15) (see Fig. 4).
Finally, despite the limit shape of the ensemble MDP(q) contains the triangle∆ q = ∆ 1/q of the same area as ∆ q , the "free" area changes to (cf. (4.23)) Moreover, according to the identity (1.10), the total area of the free parts in the limit shapes with q and 1/q is given by 1 6 π 2 − qT 2 q , which in turn implies that the total area of both limit shapes including the ground state triangles equals 1 6 π 2 , which may be interpreted as the (asymptotic) law of conservation of total energy in dual systems, that is, with limiting gaps q and 1/q. It would be interesting to find a physical explanation of this identity.  = 0.797842. The ground state triangles ∆ q and ∆ 1/q (shaded) are obtained from one another by reflection about the main coordinate diagonal. Thus, in line with Lemma 2.2, T 1/q = qT q and, equivalently, T q = q −1 T 1/q ; in particular, T 3/4 = 4 3 T 4/3 . Solid curves (red in the online version) show the limit shape graphs. According to formula (4.24), the areas under the limit shapes sum up to ζ(2) = 1 6 π 2 .
Note that equalities in (5.1) correspond to what was called the "ground state" in the discussion in Section 4.4. Now, it is natural to "subtract" the ground state by shifting the parts of λ ∈ Λ q (n, k) so as to lift the constraints (5.1) (apart from the default condition that all parts are not smaller than 1). Specifically, consider the mapping defined by ρ i := λ i + 1 − q 0 − · · · − q k−i ≥ 1 (i = 1, . . . , k).
and, recalling the notation (2.4), where n ≥ s k as long as the set Λ q (n, k) is not empty. Hence, ρ = I(λ) is a partition of the same length k and the new weight r = n + k − s k , but with no constraints on its parts; that is, ρ ∈ Λ(r, k). Moreover, it is evident that the mapping (5.2) is a bijection of Λ q (n, k) onto Λ(r, k), for each k ∈ N and any n ≥ s k . In particular, if ν q n,k is the uniform measure on Λ q (n, k) then the push-forward I * ν q n,k = ν q n,k • I −1 is the uniform measure on Λ(r, k). This observation furnishes a more straightforward way to finding the limit shape of partitions in the MDP spaces Λ q (n, k) and Λ q (n). The heuristic idea is as follows. Consider a partition λ ∈ Λ q (n, k n ), where k n ∼ τ √ n with 0 < τ < 2/q (cf. the hypothesis in Theorem 4.2). On account of the asymptotics (2.5), for the weight of ρ = I(λ) this gives In particular, k n ∼ (τ /b) √ r. Suppose now that the limit shape of ρ ∈ Λ(r, k n ) exists under the usual √ r-scaling, so that for x > 0 and r → ∞ we have approximately By the relation (5.3) and the asymptotic formulas (1.2) and (5.4), this implies which yields the limit shape for λ ∈ Λ q (n, k n ) as n → ∞. Note that the last term in (5.6) corresponds to the ground state discussed earlier, whereas the first term indicates the contribution from the "free part" of the partition λ ∈ Λ q (n, k n ). Likewise, for partitions λ ∈ Λ(n), assuming that their length follows the typical behaviour K(λ) ≈ T q ϑ −1 q √ n (see Corollary 3.9), formula (5.6) yields the limit shape where b q := 1 − 1 2 qT 2 q /ϑ 2 q (cf. (5.5)). Let us now give a more rigorous argumentation. We confine ourselves to the case q > 0 and prove a weaker statement than in the previous section (i.e., just convergence in probability instead of exponential bounds on deviations), since known results can be applied in this case. A similar approach was used by Romik [30] to find the limit shape of MDP(q) with q = 2, and by DeSalvo & Pak [11] for any positive integer q. The same technique can be worked out in the case q = 0, but this requires a more detailed analysis.
The limit shape for partitions under the uniform measure ν r,k on the space Λ(r, k) has been found by Vershik & Yakubovich [38] (see also Vershik [34]). Adapted to our notation, this result is formulated as follows. Recall that a partition ρ ′ is said to be conjugate to partition ρ ∈ Λ(r) if their Young diagrams Υ ρ and Υ ρ ′ are symmetric to one another with respect to reflection about the main diagonal of the coordinate plane. In other words, column blocks of the diagram Υ ρ become row blocks of the diagram Υ ρ ′ , and vice versa. Clearly, ρ ′ has the same weight as ρ, that is, ρ ′ ∈ Λ(r). The next result refers to the conjugate Young diagrams Υ ρ ′ , but it easily translates to the original diagrams Υ ρ . Note that the scalings used there along the two axes are both proportional to √ r but different (unless c = 1).
and y c ∈ (0, 1) is the (unique) solution of the equation Equivalently, the statement of Theorem 5.1 can be rewritten as follows: for any s 0 ∈ (0, 1] and ε > 0, Unfortunately, the condition k = c √ r + O(1) is too strong for our purposes. However, tracking the proof given in [38] and using the continuity of the expression (5.8) with respect to c, one can verify that the limits (5.7) and (5.10) hold true provided only that k ∼ c √ r.
In a similar fashion, one can prove Theorem 4.4. More specifically, by Corollary 3.9 K(λ)/k n → 1 in ν q n -probability, where k n = (T q /ϑ q ) √ n. The push-forward I * ν q n = ν q n •I −1 under the bijection I defined in (5.2) is a measure on partitions ρ ∈ Λ such that (random) r = N(ρ) and k = K(ρ) satisfy the relation r = n + k − s k . Since K(λ) = K(ρ), it follows that K(ρ)/k n → 1 in (I * ν q n )-probability. Hence, using (1.11) and (2.5), we obtain, in (I * ν q n )-probability as n → ∞, Thus, taking c = T q / Li 2 (1 − e −Tq ) it is easy to see that y c = 1 − e −Tq solves the equation (5.9). Furthermore, using (2.14) the expression (5.11) is reduced to It remains to notice, using condition (1.2), that in ν q n -probability which, together with (5.14), yields the expression ϕ Tq (t; q) for the limit shape already obtained in Theorem 4.4.

Minimal difference partitions in random environment
The assumption (1.2) that the partial sums Q k grow linearly (q > 0) or sub-linearly (q = 0) can be satisfied not just by a fixed sequence, but also by that obtained via some stochastic procedure. Similarity with random walks in random environment (see, e.g., [5] for a review and further references) allows one to speak about minimal difference partitions in random environment. Without attempting to investigate such models in full generality, we provide sufficient conditions for the basic limit (1.2) to hold with a random sequence q = (q i ). The following simple statement describes one of the possible approaches to introducing randomness into the MDP(q) context. In what follows, abbreviation "a.s." stands for "almost surely" with respect to the distribution of the environment; we denote by E the corresponding expectation. Theorem 6.1. Suppose that q = (q i ) (with the usual convention q 0 ≥ 1) is a sequence of independent integer-valued non-negative random variables, each having a finite moment of order 1 + ε for some ε > 0 and satisfying the following two conditions.