Abstract
This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever the boundary of the domain has three adjacent straight segments inclined under 120 degrees (measured in the direction internal to the domain) to each other, the asymptotics of tilings near the middle segment is described by the GUE–corners process of random matrix theory. An important step in our argument is to show that fluctuations of the height function of random tilings on essentially arbitrary simply-connected domains of diameter N have magnitude smaller than \(N^{1/2}\).
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Notes
See Lectures 11-12 in [12] for the heuristics explaining that the fluctuations should grow logarithmically in the size of the domain.
Tilings of multiply-connected regions in \(\mathbb {T}\) can give rise to height functions that are multi-valued; hence, making sense of a limit shape (see Lemma 15 below) in this context requires some additional effort. This has been done recently in [19], so our methods should also extend to the multiply-connected setting as well, but we do not address this here.
There are other definitions of the height function; see, e.g., [12, Section 1.4] for a more symmetric one. All definitions contain the same information, because they differ from each other by explicit affine transformations.
Subleading terms, i.e. \(m_N-N\cdot \left[ \lim _{N\rightarrow \infty }\tfrac{m_N}{N}\right] \) might depend on the exact way \((R_N,h_N)\) approximates \((\mathfrak R,\mathfrak h)\), and there is no way to reconstruct them only from \((\mathfrak R,\mathfrak h)\) and the corresponding tiling limit shape.
However, in some situations it fails. For instance, consider \(N\times \lfloor N^{\alpha }\rfloor \times \lfloor N^{\alpha }\rfloor \) hexagon with \(0<\alpha <1\); it obviously has an embedded trapezoid with \(I^{(l)}_N=I^{(r)}_N=\lfloor N^{\alpha }\rfloor \). For this domain, as N tends to \(\infty \), we need to rescale the positions of lozenges near the longer side by factor \(N^{1-\alpha /2}\) in order to see the GUE–corners process in the limit, as can be shown by analyzing explicit formulas of [7] and [22, Section 4]. Hence, if we rescale by \(\sqrt{N}\), as in Theorems 4 and 7, the convergence fails. One can also design more complicated examples, in which there is no way to adjust rescaling factors to restore the GUE convergence.
The points \(N\zeta _i\) in \({{\,\mathrm{dist}\,}}_{\mathcal {G}}(\cdot ,\cdot )\) need to be vertices in \(\mathbb {T}\), i.e., they should be confined to the integer lattice. Hence, we formally should write \({{\,\mathrm{dist}\,}}_{\mathcal {G}} (\lfloor N \zeta _{i - 1}\rfloor , \lfloor N \zeta _i\rfloor )\). For notational convenicence we are going to omit the integer parts throughout this proof; this will not affect the validity of the argument.
\(\lambda _{i+1}-\lambda _i\ge 1\) implies that the Lebesgue density of the limiting measure is at most one.
In (4.6) we implicitly assumed that the entire segment \((0,0)-(0, \tfrac{A+L}{N})\) lies in \(\mathfrak R\). This segment has to be inside \(R_N\) by definition, but \(\mathfrak R\) might be slightly different from \(R_N\); this can introduce at most \(o(N) = o(L)\) additional terms \(\lambda _i\) to the sum on the left side of (4.6). Since each \(\lambda _i < A + L\), the total contribution of these extra terms is at most \(o(L) \cdot \frac{1}{L} \cdot \frac{A+L}{L} = o(1)\), meaning that this difference only introduces another o(1) error, which we can ignore.
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Acknowledgements
The authors are grateful to David Keating and Ananth Sridhar for the simulation for Fig. 4. We also thank Alexander I. Bufetov for bringing the work [3] to our attention. We thank anonymous referees for helpful comments and suggestions. The work of A.A. was partially supported by a Clay Research Fellowship. The work of V.G. was partially supported by NSF Grants DMS-1664619, DMS-1949820, by BSF grant 2018248, and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison with funding from the Wisconsin Alumni Research Foundation.
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A uniformity in convergence of local statistics
A uniformity in convergence of local statistics
In this section we explain how Lemma 18 is a consequence of results from [1]. In what follows, we let \(\mathcal {B}_R (z) = \big \{ z' \in \mathbb {R}^2 : |z' - z| = R \big \}\) denote the disk of radius R centered at \(z \in \mathbb {R}^2\).
Proof of Lemma 18
As mentioned in Remark 19, Lemma 18 essentially coincides with Theorem 1.5 of [1], except that the uniformity of the constant \(C>1\) in the vertex \(v \in R_N\) was not explicitly claimed there. On the other hand, it was stated in [1] that such uniformity holds if \(R_N\) is a disk containing v. More specifically, suppose that there exists some vertex \(u \in \mathbb {T}\) such that the following holds.
-
1.
\(R_N\) approximates a disk \(\mathcal {B}_N (u)\) centered at u. That is, we have \(R_N = \mathcal {B}_N (u) \cap \mathbb {T}\).
-
2.
We have \(\mathcal {B}_{\varepsilon N} (v) \subset R_N\) (that is, \(R_N\) contains a disk centered at v).
-
3.
For each \(z \in \mathcal {B}_{\varepsilon } (\tfrac{1}{N} v)\), we have \(\nabla \mathcal {H} (z) \in \mathcal {T}_{\varepsilon }\) (that is, \(\nabla \mathcal {H}\) is uniformly liquid on that disk).
Then, Theorem 3.15 (see also Assumption 3.5) of [1] states that Lemma 18 holds, with C uniform in v. Let us mention that Theorem 3.15 of [1] does not require \(R_N\) to be tileable. If it is not, then \(H_N\) is a uniformly random height function on \(R_N\) with (some) boundary height function \(h_N\) defined on \(\partial R_N\), and
is the unique free tiling associated with \(H_N\), in which tiles are permitted to extend beyond the boundary of \(R_N\).
Although in Lemma 18 the domain \(R_N\) does not have to approximate a disk, we may apply the above result on a subdisk of it containing v in the following way. Since \(\nabla \mathcal {H} (\mathfrak {v}) \in \mathcal {T}_{\varepsilon }\), [8, Proposition 4.1] implies that \(\nabla \mathcal {H}\) is continuous in a neighborhood of \(\mathfrak {v} = \tfrac{1}{N} v \in \mathfrak {R}\). Letting \(\kappa = \kappa (\varepsilon ) > 0\) denote a sufficiently small real number to be fixed later, there then exists \(\rho = \rho (\varepsilon , \kappa , \mathfrak {R}) \in (0, \varepsilon )\) such that \(B_{\rho } (\mathfrak {v}) \subset \mathfrak {R} \cap \tfrac{1}{N} R_N\) and
Let \(\widetilde{R}_N = \mathcal {B}_{\rho N} (v) \cap \mathbb {T}\) and \(\widetilde{\mathfrak {R}}_N = \tfrac{1}{N} \widetilde{R}_N\), and condition on the restriction
of the random tiling
to \(R_N \setminus \widetilde{R}_N\). This induces (random) boundary data \(\widetilde{\mathfrak {h}}_N: \partial \widetilde{\mathfrak {R}}_N \rightarrow \mathbb {R}\) defined by setting \(\widetilde{\mathfrak {h}}_N (z) = \tfrac{1}{N} H_N (N z)\) for each \(z \in \partial \widetilde{\mathfrak {R}}_N\). Let \(\widetilde{\mathcal {H}} \in {{\,\mathrm{Adm}\,}}(\widetilde{\mathfrak {R}}_N; \widetilde{\mathfrak {h}}_N)\) denote the maximizer of \(\mathcal {E}\) on \(\widetilde{\mathfrak {R}}_N\) with boundary data \(\widetilde{\mathfrak {h}}_N\). In order to apply Theorem 3.15 of [1] on the domain \(\widetilde{\mathfrak {R}}_N\) with boundary data \(\widetilde{\mathfrak {h}}_N\), we must verify that \(\nabla \widetilde{\mathcal {H}}\) is likely uniformly liquid around \(\mathfrak {v}\) (for example, \(\nabla \widetilde{\mathcal {H}} (z) \in \mathcal {T}_{\varepsilon / 2}\) for \(z \in \mathcal {B}_{\rho / 4} (\mathfrak {v})\) with high probability).
To that end, observe for any fixed \(\delta = \delta (\varepsilon , \omega , D) > 0\) that the variational principle, Lemma 15, yields a constant \(C_0 = C_0 (\delta , \mathfrak {R}, \mathfrak {h}) > 1\) such that for \(N > C_0\) we have
Next, for sufficiently small \(\kappa = \kappa (\varepsilon ) > 0\), (A.1) and Proposition 2.13 of [1] together yield a constant \(C_1 = C_1 (\varepsilon ) > 1\) such that
Hence, if \(\delta = \delta (\varepsilon , \omega , D) > 0\) and \(\kappa = \kappa (\varepsilon )\) are chosen sufficiently small so that \(C_1 \delta < \frac{\varepsilon }{4}\) and \(\kappa < \frac{\varepsilon }{4}\), then it follows from (A.1) and (A.3) that \(\nabla \widetilde{\mathcal {H}} (z) \in \mathcal {T}_{\varepsilon / 2}\) for each \(z \in \mathcal {B}_{\rho / 4} (\mathfrak {v})\) (since \(\nabla \mathcal {H} (\mathfrak {v}) \in \mathcal {T}_{\varepsilon }\)).
In particular, Theorem 3.15 of [1] applies. Denoting \((\widetilde{s}, \widetilde{t}) = \nabla \widetilde{\mathcal {H}} (\mathfrak {v})\), it yields a constant \(C_2 = C_2 (\varepsilon , \omega , D, \mathfrak {R}, \mathfrak {h}) > 1\) such that for \(N > C_2\) we have

By (A.3), we have for sufficiently small \(\delta = \delta (\varepsilon , \omega , D) > 0\) that
Thus, by (A.2), (A.4), (A.5), and further imposing that \(\delta < \frac{\omega }{4}\), we have for \(N > \max \{ C_0, C_2 \}\) that

which implies the lemma. \(\square \)
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Aggarwal, A., Gorin, V. Gaussian unitary ensemble in random lozenge tilings. Probab. Theory Relat. Fields 184, 1139–1166 (2022). https://doi.org/10.1007/s00440-022-01168-3
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DOI: https://doi.org/10.1007/s00440-022-01168-3
Keywords
- Lozenge tilings
- GUE corners process
- Universality
Mathematics Subject Classification
- Primary 82B20 Secondary 60K35
- 60B20