# Higher-Order Airy Scaling in Deformed Dyck Paths

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## Abstract

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, ‘jumps’ orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half-length, area and number of jumps. This represents the first example of an exactly solvable two-dimensional lattice vesicle model showing a higher-order multicritical point. Applying the generalized method of steepest descents, we see that the associated two-variable scaling function is given by the logarithmic derivative of a generalized (higher-order) Airy integral.

## Keywords

Vesicles Saddle point method Basic hypergeometric series Exactly solvable models Multicritical point Phase transition## 1 Introduction

Biological vesicles act as containers transporting molecules inside cells, and consist of a lipid membrane enclosing a fluid [1]. Depending on the pressure difference between the inside and the outside, the vesicles can be found in a deflated or an inflated phase. When analysing the transition between these two phases using statistical mechanics, the pressure difference is modelled by introducing a volume fugacity into the partition function. In a two-dimensional setting, the vesicles can be described by different subclasses of two-dimensional self-avoiding lattice polygons (SAP) [2, 3], in the same way as self-avoiding walks are used to model the behaviour of long polymer chains subject to volume interactions [4]. Mathematically, the corresponding grand-canonical partition function is the generating function of the polygons, weighted with respect to their perimeter and their area.

Neglecting overhangs is also usual to consider in interface physics, for example by using a solid-on-solid model to describe the boundary of oppositely magnetized domains in the Ising model or the shape of liquid drops on a substrate at low temperatures [11, 12].

In Sect. 2, we will define DDP precisely and derive the functional equation for their generating function, weighted with respect to their area, their length and their number of jumps. An expression for the generating function in the form of a fraction of two basic hypergeometric series will be obtained in Sect. 3. The main result is given in Sect. 4, and the remaining sections contain the steps of its derivation. A contour integral representation for the series occurring in the generating function of DDP is derived in Sect. 5, which in the limit \(q\rightarrow 1^-\) has a leading contribution in the form of a saddle point integral. The location of the relevant saddle points depending on the parameters *w* and *t* is discussed in Sect. 6, and the geometry of the paths of steepest descent originating from them is investigated in Sect. 7. In Sect. 8, the integral expression for the basic hypergeometric series is then transformed into a canonical form, and the asymptotic behaviour of the coefficients of this transformation around the multicritical point is analysed. The asymptotic expression for the basic hypergeometric series is then obtained by evaluating the transformed integral in Sect. 9, which directly leads to Theorem 4.1.

## 2 The Model

The model of DDP is defined as follows.

## Definition 2.1

For \(m,s \in \mathbb {Z}_{\ge 0}\) and \(s\ge 2m\), a deformed Dyck path (DDP) of half-length *m* is a walk \((x_k,y_k)_{k=0}^{s}\) on \(\mathbb {Z}^2\), such that \((x_0,y_0)=(0,0)\), \((x_{s},y_{s})=(2m,2m)\) and \(y_k\ge x_k\) for all \(0 \le k \le s\). Moreover, if \((x_k,y_k)=(x,y)\) for \(0\le k < s\), then \((x_{k+1},y_{k+1})\) is either \((x,y+1)\) or \((x+1,y)\) or \((x-1,y+1)\), which we call an up-step, a down-step or a jump, respectively.

*k*jumps, half-length

*m*and area

*n*, with the area being defined as the number of full lattice cells enclosed between the path and the main diagonal \(x=y\). Figure 1 shows an example of a DDP with half-length 9, 3 jumps and area 12.

*G*(

*w*,

*t*,

*q*), we use the following factorization argument. A DDP has either half-length zero, or it starts with an up-step followed by a DDP followed by a down-step and then another DDP, or it starts with a jump followed by a DDP followed by a down-step followed by another DDP followed by a down-step and then another DDP—see Fig. 2 for an illustration.

*G*(

*w*,

*t*,

*q*) can therefore alternatively be interpreted as the generating function of Dyck paths, weighted with respect to their half-length and their area, with an additional weight \(F_k(w/t,1/q)\) associated to each sequence of

*k*consecutive up-steps, followed by a down-step. Here, \(F_k(s,q)\) is the generating function of appropriately weighted dimer coverings of an interval of length

*k*(for \(q=1\), see [14]).

*q*-Fibonacci polynomials \(F_k(s,q)\equiv F_k\) satisfy the recurrence

## 3 Solution of the Functional Equation

For \(q\rightarrow 1^-\), both \(\phi (-w,t,q)\) and \(\phi (-w,q t,q)\) diverge and it is therefore not immediately clear which value *G*(*w*, *t*, *q*) takes in this limit. But if we substitute \(q=1\) into Eq. (5), then we obtain a cubic equation for *G*(*w*, *t*, 1), which is readily solved. In the special case \(w=-1/9\), the radius of convergence of *G*(*w*, *t*, 1) is determined by a cubic root singularity at \(t=1/3\) and around this value we therefore expect an area-length scaling behaviour which is qualitatively different from the Airy function scaling found for Dyck and Schröder paths and staircase polygons. In order to analyse the asymptotics of the generating function in vicinity of the point \(w=-1/9\), \(t=1/3\) as \(q\rightarrow 1^-\), we apply the method of steepest descent, generalized to the case of several coalescing saddle points.

## 4 The Main Result

The principal result of this paper is stated in the following theorem, which is an immediate consequence of Proposition 9.1.

## Theorem 4.1

*s*such that \(|\Phi (s,0)|< \infty \),

The exponents \(\gamma _u\) and \(\phi _{cr}\), together with \(\gamma _t=\frac{\gamma _u}{\phi _{cr}}=\frac{1}{3}\) characterize the singular behaviour of \(G(-\frac{1}{9},t,q)\) around the multicritical point \((w,t,q)=(-\frac{1}{9},\frac{1}{3},1)\). The singular behaviour of \(G\left( -\frac{1}{9},\frac{1}{3},1-\epsilon \right) \) as \(\epsilon \rightarrow 0^+\) is determined by \(\gamma _u\); \(\gamma _t\) describes the singular behaviour of \(G(-\frac{1}{9},t,1)\) as \(t\rightarrow \frac{1}{3}^-\), and \(\phi _{cr}\) is called the crossover exponent of the model.

The critical exponents characterizing the singular behaviour of the generating function of DDP around the multicritical point, depending on the value of *w*

| \(\gamma _t\) | \(\gamma _u\) | \(\phi _{cr}\) |
---|---|---|---|

\(-\frac{1}{9}\) | \(\frac{1}{3}\) | \(\frac{1}{4}\) | \(\frac{3}{4}\) |

\({>}{-}\frac{1}{9}\) | \(\frac{1}{2}\) | \(\frac{1}{3}\) | \(\frac{2}{3}\) |

## 5 Contour Integral Representation of \(\phi (a,q^k t,q)\)

We begin by deriving a contour integral representation for the series defined in Eq. (11). The proof of the following Lemma is a straightforward modification of those given in [7] and [8] and will therefore not be carried out here.

## Lemma 5.1

*a*,

*t*with \(|\arg (1-a)|<\pi \) and \(t\ne 0\), \(0<q<1\) and \(k\in \mathbb {Z}_{\ge 0}\),

*C*is a contour from \(\infty \exp (-i\psi )\) to \(\infty \exp (i\varphi )\), with \((\psi ,\varphi )\in \,]0,\pi [^2\), intersecting the real axis at \(z=\rho \), where \(0<\rho <1\), such that all zeros of \((a/z;q)_\infty \) lie to the left of

*C*—see Fig. 5.

*z*with \(\left| \arg (1-z)\right| <\pi \), \(0<q<1\) and \(m\in \mathbb {N}\),

*n*-th Bernoulli number and \({\text {Li}}_2(z)\) denotes the principal branch of the Euler dilogarithm [6, Sect. 25.2], which for \(z \in \mathbb {C}\) is defined as

*f*(

*z*) and \(g_\epsilon (z)\) are real on the segment \(a<z<1\), therefore in this case, \(f(z^*)=f(z)^*\) and \(g_\epsilon (z^*)=g_\epsilon (z)^*\), and

*f*(

*z*) is analytic for \(z\in \mathbb {C}\setminus \big (-\infty ,a\big ]\cup \big [1,\infty \big )\). In order to analyse the integral on the rhs of Eq. (20) by means of the saddle point method, we need to further analyse the function

*f*(

*z*). We begin by discussing the location of its saddle points, depending on the values of

*a*and

*t*.

## 6 Location of the Saddle Points

*f*(

*z*) are the zeros of the derivative

*f*(

*z*) has (up to multiplicity) three saddle points \(z_i~(i=1,2,3)\), which satisfy

*t*takes one of the two values

*a*. Concerning the location of the saddle points \(z_1,z_2\) and \(z_3\) in the complex plane, we distinguish the following five cases.

- (i)
If \(a<0\), then two saddle points coalesce for \(t=t_c^+<1/4\), while the third one is negative.

- (ii)
If \(a=0\), then one saddle point is constantly zero while the other two coalesce in \(z_c=1/2\) for \(t=t_c^+=1/4\).

- (iii)
If \(0<a<1/9\), then two saddle points are mutually complex conjugates for \(0<t<t_c^-<1/3\) and coalesce on the positive real line for \(t=t_c^-\) in the point \(z_c^-\), where \(a<z_c^-<1/3\). For \(t_c^-<t<t_c^+\), all three saddle points are real and for \(t=t_c^+<1/3\), two saddle points coalesce in the point \(1/3<z_c^+<1/2\). For \(t>t_c^+\), again one saddle point is real and the other two are mutually complex conjugates.

- (iv)
If \(a=1/9\), then \(t_c^-=t_c^+=1/3\), hence all three saddle points coalesce in the same point, \(z_c^-=z_c^+=1/3\).

- (v)
If \(a>1/9\), then there is no saddle point coalescence for \(t>0\).

## 7 Geometry of the Paths of Steepest Descent

In this section we are going to discuss the geometry of the paths of steepest descent of *f*(*z*) (see e.g. [20] for a general introduction to the method of steepest descents). To this purpose, we first state the following Lemma, the proof of which relies on basic relations for the Euler dilogarithm [6, Eq. 25.12.4].

## Lemma 7.1

*b*denotes the principal branch value of \(\ln (a t)\) and \(\ln (t)\) with \(\ln (-1)=i\pi \) for \(\lambda \rightarrow 0^+\) and \(\lambda \rightarrow \infty \), respectively, and \(\psi =2\,\phi \mp \pi \) for \(\phi \gtrless 0\).

*x*, real \(t>0\) and \(0<a<1\),

Using Lemma 7.1, we prove

## Lemma 7.2

## Proof

*f*(

*z*) stated in Lemma 7.1, we can conclude that paths of steepest descent can only end in \(z=0\) or at \(\infty \exp (\pm i\pi /2)\). Since \(f(z^*)=f(z)^*\), it is sufficient to consider the upper half-plane. There are two cases to be distinguished.

- 1.
\({0< t < t_c^-(a)}\). In this case, \(z_3\) is real while \({\text {Im}}(z_1)>0\). One of the two paths of steepest descents originating from \(z_1\) ends in \(z=0\), while the other one ends at infinity. Since paths of steepest descent can only cross in saddle points, it follows that the path of steepest descent emerging from \(z_3\) necessarily ends at \(\infty \exp (i\pi /2)\). Figure 7a shows an example for this case.

- 2.
\({t_c^-(a) \le t \le t_c^+(a)}\). In this case, all three saddle points are real. The path of steepest descent originating from \(z_1\) necessarily ends at zero, while the path of steepest ascent originating from \(z_2\) ends in the point \(z=-t\). Again it follows that the path of steepest descent originating from \(z_3\) ends at \(\infty \exp (i\pi /2)\). Figure 7b shows an example for \(t_c^-(a)< t < t_c^+(a)\) for \(a<1/9\) and (c) shows the special case \(a=1/9\), for which the three saddle points coalesce.

For \(0 < t\le t_c^+(a)\), \({\text {Im}} f(z_3) = 0\). Since the paths of steepest descent are the contours on which the imaginary part of *f*(*z*) is constant, the union of the two paths of steepest originating from \(z_3\) and ending at \(\infty \exp (\pm i\pi /2)\) has the properties of the curve \(c(\lambda )\). \(\square \)

## 8 Transformation of *f*(*z*) into a Canonical Form

*f*(

*z*) coalesce in the point \(z_c=1/3\) for \(t=t_c^+(1/9)=1/3\). We now define the natural coordinates

*f*(

*z*) and \(g_\epsilon (z)\) as functions of

*z*, \(\tau \) and \(\delta \) from now on.

*p*(

*u*) by \(u_1,u_2\) and \(u_3\), hence

*f*(

*z*) need to be mapped onto the saddle points of

*p*(

*u*). For \(\tau =\delta =0\), the three saddle points of

*f*(

*z*) coalesce and it follows from Eq. (36a) that \(u_1=u_2=u_3=0\). With this we obtain from Eq. (36b–c) that \(\alpha _{0,0}=\beta _{0,0}=0\).

*p*(

*u*) such that \(u(z_j) = u_{j}\) for \(j=1,2\) and 3, then from Eq. (33) it follows by differentiating twice that

### 8.1 Coefficient Asymptotics for \(\tau \rightarrow 0\) and \(\delta =0\)

From the above discussion we know that for \(\delta =0\) and \(\tau \rightarrow 0\), \(\alpha \sim \alpha _{r_\alpha ,0}\tau ^{r_\alpha }\) and \(\beta \sim \beta _{r_\beta ,0}\tau ^{r_\beta }\), where \(r_\alpha ,r_\beta \in \mathbb {N}\) and \(\alpha _{r_\alpha ,0},\beta _{r_\beta ,0}\ne 0\).

*u*and insert the saddle point values. This gives us

### 8.2 Coefficient Asymptotics for \(\delta \rightarrow 0\) and \(\tau =0\)

It follows from an argument analogous to the one given in the previous subsection that for \(\tau =0\) and \(\delta \rightarrow 0\), \(\alpha \sim \alpha _{0,1}\delta \) and \(\beta \sim \beta _{0,1} \delta \), where \(\alpha _{0,1},\beta _{0,1}\ne 0\).

## 9 Asymptotics of \(\phi (a,q^k t,q)\)

It follows from Lemma 7.2 together with Cauchy’s theorem that for \(0<t\le 1/3\) and \(a\le 1/9\), we can replace the integration contour in Eq. (20) by a contour \(C_0\) originating from \(\infty \exp (-i\frac{\pi }{2})\), passing through the real valued saddle point \(z_3\) of *f*(*z*) and ending at \(\infty \exp (i\frac{\pi }{2})\), such that \({\text {Im}} f(z)=0\) on this contour and \({\text {Re}}f(z)\) is maximal at \(z_3\).

*p*(

*u*) ending at \(\infty \exp (\pm i\pi /4)\). Extending the integration to the complete contour, we obtain

*u*thereof at \(u=0\) for \(\tau =\delta =0\) gives together with (38) for \(\tau =\delta =0\),

## Proposition 9.1

*a*and

*t*and \(A(a,q) = (q;q)_\infty (a;q)_\infty \).

*G*(

*w*,

*t*,

*q*) around the critical point \((w,t,q)=(-\frac{1}{9},\frac{1}{3},1)\) as stated in Theorem 4.1. In Fig. 8, we show the convergence of the asymptotic approximation of

*F*(

*s*) obtained by rearranging Eq. (14) against the exact scaling function.

As discussed in Lemma 7.2, for \(a<\frac{1}{9}\) and \(0<t<t_c^+(a)\), the integration contour *C* used in Eq. (15) can be deformed such that it consists of two paths of steepest descent, connecting a saddle point on the real axis with infinity, and the asymptotics of \(\phi (a,q^k t,q)\) can be obtained via the ordinary method of steepest descent. According to Sect. 6, the relevant saddle point coalesces with another saddle point for \(t=t_c^+(a)<\frac{1}{3}\). At this point, \(\phi (a,q^kt,q)\) can be approximated in terms of Airy functions, with the special case \(a=0\) having been treated in [8].

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