Generalized multicritical one-matrix models

We show that there exists a simple generalization of Kazakov’s multicritical one-matrix model, which interpolates between the various multicritical points of the model. The associated multicritical potential takes the form of a power series with a heavy tail, leading to a cut of the potential and its derivative at the real axis, and reduces to a polynomial at Kazakov’s multicritical points. From the combinatorial point of view the generalized model allows polygons of arbitrary large degrees (or vertices of arbitrary large degree, when considering the dual graphs), and it is the weight assigned to these large order polygons which brings about the interpolation between the multicritical points in the one-matrix model.


Introduction
Matrix models have been among the most important tools when discussing non-critical strings or 2d quantum gravity coupled to conformal field theories with central charge c < 1.The main interest in the gravitational aspect came from attempts to non-perturbatively regularize the Polyakov path integral in spacetime dimension different from 26 [1,2,3,4].While the stringy aspect of this program partly failed for physical target space dimensions, the 2d gravity aspect was a very fruitful area of research, initiated in [5,3], and getting full attention after the seminal paper [6] by Kazakov.The latter used the Hermitian matrix model in the large N limit to describe certain matter fields interacting with 2d quantum gravity.Eventually it was understood that the models in [6] describe 2d quantum gravity coupled to (2, 2m − 1) conformal field theories, m = 2, 3, . . .[7] (see e.g.[8] for a review).The susceptibility exponents of these theories were calculated (in a way we will discuss below) to be given by To obtain exponents corresponding to other conformal field theories one had to consider multi-matrix models [9,10,11].In this paper we will show that one can in fact obtain the full range of exponents γ s ∈ ]−∞, 0[, in the large-N limit of the standard one-cut Hermitian matrix model by allowing for potentials with "heavy tails".In the range s ∈ ]3/2, 5/2[ these matrix models have a combinatorial interpretation in terms of random plane graphs (or random planar maps) with high degree vertices or polygon, which have been of recent interest in the mathematical (physics) literature [12,13,14].The rest of this paper is organized as follows.In Sec. 2 we remind the reader of the multicritical matrix model introduced in [6].In Sec. 3 we generalize the results of Sec. 2, such that any critical exponent γ s < 0 can occur.The corresponding potential V s (x) as well as its derivative V s (x) are infinite power series in x with cuts on the real axis.We suggest how one can associate a central charge c(s) to each s.In Sec. 4 we show that the standard way of solving the saddle point equation is still valid.Next we address the question of universality (Sec.5) and the corresponding continuum limit (Sec.6).The generalized Kazakov potentials V s (x) where s ∈ ]1, ∞], allow for a combinatorial interpretation which will be described in Sec.7 and the relation to O(n) models on random triangulations is outlined in Sec. 8. Finally Sec. 9 summarizes our results.

The Multicritical matrix model
Let us consider the following N × N Hermitian matrix model where In the large-N limit there is a one-cut solution, where the eigenvalues of M condense in an interval [−a, a] and the so-called resolvent (also called the disk amplitude) is an analytic function of z outside the cut.The large-N solution for W (z) is where the condition W (z) → 1/z for |z| → ∞ implies , B(x, y) = Γ(x)Γ(y) Γ(x + y) .
This fixes g(a 2 ) as a polynomial of a 2 for a polynomial potential.We can rewrite the integral representation (5) for W (z) in terms of the function g(a 2 ) instead of the potential Ṽ (x).Let us introduce a special notation for this function so the boundary equation (6) reads Ũ (a 2 ) = g(a 2 ).(8) Then one has the following representation of W (z) The proof is based on the identity 2 ) For a general potential defined by a convergent power series we have from ( 6) the relation Finally note that (8) and ( 9) lead immediately to the known equation for the disk amplitude with one puncture: A so-called m th multicritical point of this matrix model is a point where In order to satisfy this requirement an even potential Ṽ (x) has to be at least of order 2m.If we restrict ourselves to potentials of this order1 g(a 2) is fixed to be of the form The value a 2 c > 0 can be chosen arbitrary, after which the coefficients v n are completely fixed.For convenience we choose a c = 1, i.e.
From ( 6) and ( 15) we obtain the coefficients v n (m) for the m th multicritical Kazakov potential where the last equality should be understood as the limit where m goes to an integer.For future use we write the m th Kazakov potential as 3 The generalized Kazakov potential Let us now generalize the potential (17) by simply allowing s in v n (s) to be a real number larger than 1/2.We thus introduce where 3 F 2 is the generalized hypergeometric function.Formally, taking s → m + 1/2 the infinite sum is automatically terminated at n = m, and the m th multicritical Kazakov potential is reproduced.For s = m + 1/2 the coefficients behave as v n (s) ∼ n −s−1 for n → ∞ and therefore Ṽs (x) is a power series with radius of convergence equal to one.Given (18) we find and further, from (11): and which is the most obvious generalization of ( 14).
If we formally apply ( 9) we find for the potential ( 18) where the relation between a and g is given by ( 21), i.e.
All the representations of W (z) given above have their virtues as we will now describe.
A standard representation of gW (z) for an ordinary (even) polynomial Ṽ (z) of degree 2n is where M (x) is a polynomial of degree n − 1, uniquely fixed to cancel Ṽ (z) and to insure that W (z) → 1/z for |z| → ∞.In our case Ṽ (z) will have a cut along the real axis starting at z 2 = 1 as is clear from (19).Correspondingly M (z 2 − a 2 ) should thus have a similar cut and ( 23) is simply the representation (27) and we have (28) from which we can read off the coefficients M k .
The representation (24) is useful because the hypergeometric functions are analytic along the cut z ∈ [−a, a] of W (z), 0 < a < 1, and thus the discontinuity across the cut is entirely determined simply by the discontinuity of √ z 2 − a 2 .From the very definition (4) of W (z) it follows that the density of eigenvalues, ρ(x), is determined by the discontinuity of W (z) across the cut and we thus obtain: This ρ(x) is plotted as a2 → 1 (or g → g * ) in Fig. 1 for several values of s.Up to normalization these plots correspond to (1 − x 2 ) s−1 since we can rewrite ρ(x) as where the part in brackets is bounded for all a ∈ [x, 1] for fixed x 2 < 1.Further, ρ(x) is positive in x ∈ ]−a, a[, vanishes at x = ±a and tends to the delta-function as s → ∞.
Finally the representation (25) shows that W (z) indeed has convergent power expansion in 1/z for |z| sufficiently large and using (20) it follows that W (z) → 1/z for |z| → ∞.For future reference we note that the transformation of the hypergeometric functions from (24) to (25) involves terms not seen in (25).More specifically one has but the last term on the rhs of eq.(31) cancels against an identical term coming from the other hypergeometric function in (25).Let us end this section by calculating the susceptibility exponent γ s associated with the matrix model with potential (18).We define the susceptibility as the second derivative of the free energy of the matrix model with respect to the coupling constant g: and γ s by where χ a (g) is analytic at g * .Expanding d(gW (z))/dg in inverse powers of z, any of the terms c n (g)/z 2n+1 , n > 1, will have (g * − g) −γs as the leading non-analytic term.From (12) and (26) it follows immediately that the term is (g * − g) 1/(s−1/2) and therefore (34) For s ∈ ]m − 1/2, m + 1/2[ with m a positive integer our potential (18) has many of the characteristics of the s = m + 1/2 multicritical potential: the first m terms in the power series have oscillating signs, starting out always with x 2 /2.The signs of terms x 2n , n ≥ m are the same.At the same time, moving s towards m + 1/2, γ s changes continuously towards the value −1/m of the m th multicritical model.The range s ∈ ]1/2, 3/2] is special.It starts out with s = 3/2, i.e. m = 1 and thus Ṽ (x) = x 2 /2, i.e. a trivial Gaussian potential and we have gW a 2 is an analytic function of g, in accordance with the value γ s = −1.For 1/2 < s < 3/2 all coefficients in the power series expansion of Ṽ (x) are positive and the derivative g (a 2 = 1) is infinite rather than zero as for s > 3/2.For s For the m th multicritical potential it is well known that γ = −1/m does not correspond to the KPZ area susceptibility exponent γ A [7].Rather, it is related to insertions of the primary operator with the most negative scaling dimension, which in non-unitary conformal theories coupled to 2d gravity need not be the cosmological constant.In the multicritical models one obtains the KPZ exponent by identifying the cosmological constant via the length of the boundary of the disk.One thus looks at where the average • is with respect to the partition function (2).We are interested in the limit → ∞ where the integral will be dominated by x close to the boundary a.One obtains the leading behavior where we have used (26).Thus we identify the dimensionless boundary cosmological constant µ B and we introduce the dimensionless bulk cosmological constant µ ∼ µ 2 B as follows From the definition (32) we have and we conclude that If we assume that γ A is related to an underlying conformal field theory coupled to 2d quantum gravity, as is the case for the multicritical points where s = m+1/2, we have from the standard KPZ relation that the central charge of the matter fields related to s is The same c(s) corresponds to two different γ A 's, related by The two γ A 's correspond to the two different solutions to the KPZ relation (41).Usually the conformal field theory associated with a given central charge c is assigned a γ(c) from the branch where γ(c) → −∞ for c → −∞.However, the other branch also has an interpretation in terms of random surfaces and 2d quantum gravity [17,18,19].
If we follow the above conjectures we are led to the following picture: s = 2 corresponds to c = 1 (γ A = 0) where the two branches meet.The region s ∈ ]2, ∞[ corresponds to the "physical" branch of the KPZ equation where γ A changes from 0 to −∞.The other branch corresponds to s ∈ ]1, 2[ and γ A > 0, approaching 1 for s → 1 + .An interesting example is s = 3/2 considered above.Formally it corresponds to the m = 1 "multicritical" matrix model which is just the Gaussian matrix model with W (z) given by (35).In KPZ context it can be viewed as the (2,1) conformal field theory coupled to 2d gravity in the series of (2, 2m − 1) conformal field theories corresponding to the multicritical models, although it, contrary to the larger m theories, is not a standard minimal conformal field theory.The KPZ assignment of central charge to this theory is c = −2 and the corresponding γ A = −1.In fact we found γ s = −1 above, but according to (40) the corresponding γ A = 1/2, in agreement with the fact that W (z) in (35) is the partition function for branched polymers which is known to have γ = 1/2.That branched polymers play an important role in the interpretation of γ A is the essence of the work [17,18,19].It also follows from (42) that s = 3/2 → s = 3 and s = 3 indeed gives c = −2 and γ A = −1.In Sec. 7 we will see it is possible to give a combinatorial explanation of the relation between s and s which is in agreement with the picture picture developed in [17,18,19].
Clearly s = 1 is special, being the limit where the assumed central charge c(s) → −∞ and γ A → 1.The potential (18) is in this case and the corresponding disk function from ( 25) It is interesting that all potentials corresponding to integer s > 1, i.e. non-negative integer γ A , are simple modifications of (43).Similarly the corresponding W s are simple modifications of (44).These statements follow from Gauss' recursion relations for hypergeometic functions.

The Riemann-Hilbert method at work
Above we assumed that one can use the standard large N one-matrix model formula to obtain the disk function.Let us briefly discuss why the formula is still valid in certain cases where V (x) has cuts and poles at the real axis.It represents a simple generalization of the usual case of the one-matrix model with polynomial V (x) which can still be treated by the Riemann-Hilbert method.
The large N saddelpoint of the matrix model is the principle value integral which is valid when x belongs to the support of the eigenvalue density ρ which is assumed to avoid possible cuts and poles of V .We proceed in the usual way by introducing the analytic function and rewriting eq. ( 45) at the real axis as Usually, the term with V is missing since V is real at the real axis, but we now have to include it since V can have cuts located on the real axis.Equation (47) on the real axis implies the following equation in the whole complex plane: where the contour C 2 encircles possible cuts and poles of V (ω) on the real axis, but not z and not the cut(s) of W (ω). Q(z) is an entire function (a polynomial if V is itself a polynomial) and its role is to compensate nonnegative powers of z in the product V (z)W (z).The third term on the left-hand side of eq. ( 48) plays thus no role in determining Q(z).
We can rewrite eq.(48) as where the contour C 1 encircles (anti-clockwise) the cut(s) of W (ω), but not z and possible cuts and poles of V (ω).We can prove the equivalence of Eqs. ( 48) and (49) by deforming the contour C 1 in eq.(49) to C 2 , which will give the third term on the left-hand side of eq. ( 48).We get in addition the residual at ω = z, which accounts for the second term on the left-hand side of eq. ( 48), and finally we get the contribution from ω = ∞, which is equal Q(z).Equation ( 49) is the usual loop equation of the one-matrix model at N = ∞ with the potential tr V (M ).Its standard derivation by an infinitesimal shift of M apparently works for all potentials, including the ones with cuts on the real axis.Correspondingly, eq. ( 49) results in the usual formula for the one-cut solution where the cut is from a to b.For an even potential V (x) = V (−x), when the cut is from −a to +a, it simplifies to (5).The values of a and b are determined from the condition W (z) → 1 as z → ∞, which for an even potential reduces to (6).Explicit formulas for a simplest non-even logarithmic potential are presented in Appendix A.

Universality
Let us recall the universality situation when the potential V (x) is (an even) polynomial.
Using a Wilsonian wording we have an infinite dimensional space of coupling constants, the coefficients in all polynomials V (x) and the m th critical surface is characterized by the condition (13).It has finite co-dimension m − 1 and one can approach the surface such that m − 1 parameters survive in the "continuum" limit (see [20] for a review).The Kazakov potential ( 17) is a particular simple choice of polynomial which only depends on one parameter, g.We would like to understand the universality situation for the new critical points defined by the generalized Kazakov potentials V s (x) = 1 g Ṽs (x).Clearly the new critical behavior is related to the tail v n ∼ n −1−s in Ṽ (x).Let us choose another potential with the same tail but depending on two parameters, g and c, rather than the single g in V s (x), where Li 1+s is the polylogarithm.This potential is rather general.In particular, we can get a quartic potential from (51) in the limit c → ∞, g ∼ 1/c.The boundary equation ( 8) now reads where the function F s (A) (trivally related to Ũ (A)) is defined by It has the following expansion (see also (11)) Using the properties of F s (A) listed in Appendix B, one can analyse the function g(a 2 ).It is an analytic function of a 2 for 0 ≤ a < 1 and the behavior close to a 2 = 1 is as follows: where f (x) can be expanded to order [s − 1/2]: Figure 2: Value of c * versus s above which the usual 2d gravity scaling limit is realized for the polylog potential (51).
The function g(a 2 ) starts out as an increasing function of a 2 .By increasing a, eventually a might become a non-analytic function of g.This happens either at the first a where g (a) = 0 or, if g (a) > 0 for all a, at a = 1, the radius of convergence for g(a 2 ).In the former case we have an a c < 1 where g (a c ) = 0 and a corresponding critical value of g, g c = g(a c ).For our choice of the potential (depending only on g, c) one can show that g (a) = 0 for all values of a < 1.
In a neighborhood of a c we can therefore write We thus conclude that the leading non-analytic behavior of a as a function of g is (a 2 c −a 2 ) 1/2 , i.e. we have the standard situation with γ s = −1/2, corresponding to the m = 2 Kazakov potential.Whether or not this situation is realized depends on the value of c.We have and thus the following equation for the value c * (s) separating the two situations: This c * (s) is positive for s > 3/2, because then 1 < F s−1 (1) < ∞ and increases rapidly with s as is depicted in Fig. 2. Let us first discuss the situation for s ∈ ]3/2, 5/2[.For a given s in this interval and a given c ≤ c * (s) the critical point is thus g * s corresponding to a c = 1 and the relation between a and g close to a c is determined by (55) and (56).For fixed c < c * (s) the analytic term from f (1 − a 2 ) will dominate over the non-analytic term (1 − a 2 ) s−1/2 and we have formally the situation corresponding to γ = −1.However, precisely for c = c * (s) this term will by definition vanish and we obtain from (55) i.e. precisely the same scaling relation as for the generalized Kazakov potential, and thus also γ s = 1/(1/2 − s).If c > c * (s) we have γ s = −1/2, but for c → c * (s) (60) will take over since the term non-analytic in (1 − a 2 ) will dominate over the contribution (57) when a c → 1.In the limit s → 5/2 they will agree and give γ 5/2 = −1/2.If we consider s ∈ ]5/2, 7/2[ we still have the same the curve c * (s), and results identical to those for s ∈ ]3/2, 5/2[ if c = c * (s).For c = c * (s) the term in f (1 − a 2 ) linear in (1 − a 2 ) will still cancel, but the term proportional to (1−a 2 ) 2 will be dominant compared to (1−a 2 ) s−1/2 .Only if we can cancel the analytic (1 − a 2 ) 2 term will we obtain a scaling like (60) also for s ∈ ]5/2, 7/2[.To obtain such a cancellation we need one further adjustable coupling constant apart from g and c.
There are many ways to introduce such a coupling constant, but maybe the simplest is to add a term v 2 x 4 to the potential (51).With this new coupling constant at our disposal we can always find a point a c < 1 such that g (a 2 c ) = 0. We can also try to find a point a c where not only g (a 2 c ) = 0 but also g (a 2 c ) = 0, precisely as for the m = 3 multicritical matrix model.In Fig. 3 we show such a situation.Whether or not this is possible depends again on c and corresponding to eq. ( 59) one obtains and the corresponding value of v 2 (s) is We show c * (s) and v 2 (s) in Fig. 4. Note that they are both negative.For c < c * (s) we can approach c * (s) by changing c while satisfying g (a c ) = g (a c ) = 0, where a c (c) < 1 and a c (c) → 1 for c → c * (s).The condition g (a c ) = g (a c ) = 0 determines v 2 uniquely for fixed c.For c > c * (s) one can approach c * (s) in such a way that g (a c ) = 0.This does not fix v 2 and the corresponding a c , but by demanding that v 2 → v 2 (s) given by (62) we have by construction that a c → 1 and g (a c ) → 0 for c → c * (s).For s ∈ ]5/2, 7/2[ we thus have a situation completely analogous to s ∈ ]3/2, 5/2[, except that the multicriticality while approaching c * (s) has changed from m = 1 and 2 to m = 2 and 3.At c * (s) we have γ s = −1/(s−1/2) and the potential V (x) is qualitatively the same as the generalized Kazakov potential V s (x) in the same range of s.
The generalization to higher values of s is straight forward.For s ∈ ]m − 1/2, m + 1/2[ we allow deformations of V involving v 2 , . . ., v m−1 .We can define a critical c * (s) and approach it from the two sides via m − 1 and m critical points by changing c, and the potential V (x) at c = c * (s) will be qualitatively the same as V s (x).We have thus seen that the new scaling limits for s > 3/2 are universal in the same way as the standard multicritical points of the one-matrix model which correspond to s = m + 1/2.
Let us finally consider the region s ∈ ]1/2, 3/2[.For s in this region we have Thus a c = 1 and γ s = −1/(s − 1/2) and no fine tuning of c is needed (except if one insists on g * s positive one has to choose c negative).The range of γ s is from −1 to −∞, i.e. outside the range of the original Kazakov range of γ(m) = −1/m with integer m.

The continuum limit
For the (even) matrix models the scaling limit is usually performed by the following assignment In our case a c = 1.For most of the "observables" considered for matrix models, this scaling is straight forward and unproblematic.As examples we have for the disk amplitude with one puncture, d(gW (z))/dg, that and for the universal two-loop function (which can be derived for our more general potentials precisely as for the ordinary polynomial potentials [21,22]): The same is true for any higher loop functions.Approaching the m th multicritical point for ordinary matrix models one obtains where W cont (P 1 , . . ., P b ; Λ) denotes the continuum b-loop function2 .The one (natural) difference in our more general case will be that in the divergent pre-factor m is replaced by s − 1/2.The so-called continuum limit of the disk amplitude requires a more detailed discussion since it contains a non-scaling part.If we use the representation (27) the potential term Ṽ (z) will not scale when using the prescription (64).On the other hand the rest of the expression will scale, as is clear from (27) for a polynomial potential and from ( 28) for the generalized Kazakov potential.However the rhs of (28) does not fall off as a function of the continuum P for |P | → ∞ the way one requires for the continuum disk-amplitude W (P ).One cures this by introducing a "continuum" potential V cont (P ) which is determined by the requirement3 that W cont (P ) has a power expansion in P −n− 1 2 , n ≥ 0, for P → ∞.We thus write4 That the scaling factor is s−1 follows immediately from (28).When s = m + 1/2 it reduces to the ordinary scaling factor for the ordinary m th multicritical matrix model.Equations ( 25), ( 28) and (31) and the remarks surrounding (31) allow us immediately to substitute the continuum limit (64) and we obtain This W cont (P ) can indeed be expanded in powers 1/P n+ 1 2 and the series is absolutely convergent for |P | > √ Λ and from the integral representation of hypergeometric functions it follows that it is analytic for positive P .It has a cut for negative P starting at P = − √ Λ, coming from P + √ Λ.Like for the ordinary matrix models, this cut is the scaled version of the original cut [−a, a] in z.The potential Ṽ cont (P ) in (71) has a cut along the positive P axis.This is the scaled version of the original cut of Ṽ (z) starting at z = 1.
If s = m + 1/2 it is instructive to rederive the standard continuum results for the m th model directly from (28).Using (28) we obtain where the expression part in square brackets is a polynomial in P of order m − 1, which can be written as c k being the coefficients in the Taylor expansion of 1/ √ 1 − x.This implies that except for P m−1/2 all positive powers of P will cancel on the rhs of eq. ( 72) and we obtain i.e. precisely the representation (69)-( 71).Let us briefly discuss the perturbation away from one of the generalized multicitical points.One convenient way to characterize the deformation away from the ordinary m th multicritical point is to use the so-called moments M k [21,22,23].They are defined by 5 In (75) we view Ũ and M k as functions of a 2 and the coupling constants v n .For a given choice of coupling constants v n and g the position or the cut, i.e. the determination of a as a function of v n and g will then finally be determined by (8).The coupling constants v c n and g * correspond to an m th multicritical point if the corresponding value a = a c is such that In the case of the Kazakov potential we have chosen a particular simple way to move away from the critical point, namely by keeping the v n = v c n and only changing g and that case we had explicitly For the generalized Kazakov potential this is changed to 5 An equivalent definition is where the contour C encircles to cut of W (z) but not any poles or cuts of V (x).
the difference being that now infinitely many moments are different from zero.
For the m th multicritical model a general deformation away from the multicritical point could be described as a change of coupling constants away from the critical values such that where µ k and √ Λ are kept fixed when the coupling constants change towards their critical values.As shown in [23] all multiloop functions can in the continuum limit be expressed as functions of µ k 's, √ Λ and the variables P 1 , . . ., P b .The obvious generalization to a deformation around the generalized Kazakov potential is to assume that and that the µ k 's and Λ are kept fixed when the coupling constants flow towards their critical values.With such a behavior all formulas for multiloop functions derived for the deformation around an arbitrary m th model will remain valid of any choice of s.For an arbitrary s > 1/2 it is possible to define so-called continuous times T k , related to the µ k 's, and to study the so-called KdV flow equations in terms of the T k 's.Details of this will appear in a forthcoming paper [24].

Combinatorial interpretation
To better understand the duality s → s s−1 discussed in Sec. 3 let us have a look at the combinatorial interpretation of the matrix model in terms of planar maps, i.e. graphs embedded in the plane modulo orientation-preserving homeomorphisms.The boundary of a planar map m is the contour of its "outer face", and we assume that m has a distinguished oriented edge on the boundary, which is called the root edge.We denote by M (l) , l ≥ 1, the set of all such rooted planar maps that are bipartite, i.e. having all faces of even degree, and have boundary length 2l.By convention we let M (0) contain a single map consisting of just a vertex.If we write q n := δ n,1 − 2n v n for n ≥ 1 then the disk amplitude W (z) for z 2 ≥ a 2 (g) and g ≤ g * can be expressed as the convergent sum One should notice that M (l) contains planar maps with a boundary of the most general "non-simple" form, meaning that it may have pinch points in the sense that vertices appear multiple times in the boundary contour (see figure 5 for an example).As we will see shortly, if s ≤ 2 dropping the contribution of planar maps with non-simple boundaries from (80) has a non-trivial effect on the scaling properties of the disk amplitude.
We denote by M(l) ⊂ M (l) the planar maps with a "simple" boundary, meaning that all vertices in the boundary contour are unique, and define the simple disk amplitude Ŵ (x) for x 2 sufficiently small by From the dual point of view W (z) and Ŵ (x) can be interpreted respectively as the disconnected and connected planar Green functions, and it has long been recognized that they satisfy a simple relation [15,16].Indeed, since a planar map m with non-simple boundary contains a unique submap with simple boundary sharing the same root edge (see figure 5), one easily observes that This implies that Ŵ (x) = gx W −1 (x) when |x| ≤ W (a(g)), where W −1 (•) is the functional inverse of z → W (z). Notice that the position of the cut in this simple disk amplitude is now determined by W (a) which when a → 1 scales as Which of the two last terms dominates depends on whether s > 2 or s < 2 (we will not discuss integer s).In particular, if one identifies the "simple" boundary cosmological constant μB in analogy with the discussion above (38) one obtains μB ∼ (1 − g/g * ) which is invariant under s → s/(s − 1) and corresponds to the "right" branch of (41).
Let us now have a look at the continuum limit of the simple disk amplitude using (69).Based on (85) one expects that one should scale x 2 → x 2 c (1 − X β ) with x c = (s − 1/2)/(s − 1) and β = 1 for s > 2 and β = s − 1 for s < 2, in addition to a 2 → 1 − √ Λ .If we denote the leading order of W −1 (x) in by W −1 (x) ∼ 1 + P /2 then for P > 0 It follows that for s > 2 we have which has the same form (up to rescaling) as the continuum limit of the non-simple disk function W (z). On the other hand, when s < 2 one may check that W Λ (P ) is monotonically decreasing on P ∈ [− √ Λ, ∞[ and therefore we identify Since we took x 2 → x 2 c (1−X s−1 ), the linear term in (92) is in fact the dominant singular part and we conclude that it is really the functional inverse of (90) that provides the continuum limit of the simple disk amplitude.

Relation to the multicritical O(n) loop models
The range of universality classes parametrized by s ∈ ]1, ∞[ is akin to that of the multi-critical O(n) models studied in [11].This is not a coincidence: it has been observed in [12,13] that at criticality there exists a natural relation between O(n) models and random planar maps with non-trivial weights on the faces.Let us briefly describe this connection.The O(n) matrix model for positive integer n is defined by where k qk M k is some polynomial potential and z * is an independent coupling constant.The corresponding Feynman diagrams in the large N limit can be interpreted as the duals of loop-decorated planar maps, as shown in figure 6 which makes sense for any real value of n.It is shown in [11] that for n ∈ [−2, 2] one may tune the parameters qk , k = 2, . . ., m+1, together with z * such that the disk amplitude W (z) takes the form Of course, when n = 0, i.e. b = 1/2, the loops are suppressed and one is back at the standard multi-critical matrix model (although with a different potential than the one in Sec. 2 since it is not restricted to be even).
To better understand the connection between the two models it is convenient (see also [13]) to introduce the gasket G(m) of a loop-decorated planar map m with a boundary to be the planar map obtained by removing all triangles intersected or surrounded by loops (see figure 6).Given a planar map m with boundary (and no loops), one may ask for the total weight in the sense of (94) of all loop-decorated planar maps m that have m = G(m) as their gasket.Since each face of m corresponds to either a face or a loop of m, we easily find that this total weight factorizes as f ∈Faces(m ) q deg(f ) where q k is given by This is precisely the weight associated to m in a one-matrix model with "effective" potential and therefore W (z) may be identified as the disk amplitude of this one-matrix model.In particular the singular behavior V eff (z)| sing.∼ (z * − z) m−b agrees with that of ( 18) when s = m − b + 1.
The precise connection with the multi-critical O(n) model only holds at criticality, i.e. g = g * .This explains why the continuum limit (70) of our disk amplitude for Λ = 0 is quite different from the standard one of the O(n) model which reads [11] W Note that one can obtain a connection away from criticality, i.e. g < g * , if one is willing to supplement the O(n) model weight (94) with a factor g/g * for each vertex in the gasket, i.e. for each vertex not surrounded by a loop.

Conclusions
We have shown that standard matrix model calculations extend to potentials of the form Both the potential and their derivatives have cuts on the real axis.Nevertheless one can find 1-cut solutions W (z) to the disk amplitude which are natural generalizations of the standard multicritical disk amplitudes and in this way the generalized Kazakov potentials V s (x) serve as generalized multicritical points interpolating between the standard multicritical points.In particular the b-loop functions are universal functions when expressed in terms of z 2 j − a 2 , j = 1, . . ., b and the b − 2 first moments M k , k = 1, . . ., b − 2, even if W (z) itself depends on infinite many M k 's.Also, for the multiloop functions the continuum limit is obtained in a straigth forward manner.
To each s > 1 one can formally associate a central change c(s) given by (41) and conversely to each central charge c < 1 one can associate two values s(c) > 2 and s (c) < 2 related by s = s/(s − 1) corresponding the KPZ exponents γ A (s) = 2 − s and γ A (s ) = 2 − s , related by (42) and corresponding to the two solutions of the KPZ equation (41).The "wrong" solution of the KPZ equation where γ A (s ) > 0 has been associated with so-called touching interactions where one in matrix model context has added terms like g t (trφ 2 ) 2 to the ordinary matrix potential.By fine-tuning the touching coupling like g t one could obtain certain critical exponents > 0. We have here seen very explicitly in Sec.7 that for potentials with the most heavy tail, namely 1 < s < 2 the "touching" picture appears automatically, without adding any explicit touching interaction, and that the whole range 0 < γ A < 1 is spanned.
A number of interesting questions remain to be answered.Is there any conformal field theory interpretation of the region 1/2 < s < 1? How do the perturbations away from the generalized Kazakov point relate to the corresponding conformal field theory, which in general will be irrational?What is the most natural way to perturb away from the generalized Kazakov point and how does it relate to the standard KdV flow equations valid for any standard multicritical model?These questions deserve further considerations.
The asymptotic behavior near A = 1 can be found from the difference If we expand the difference in (1 − A), we find where the integral converges for s > 1 + α.We then find (122) Thus the asymptote (121) holds for α < s < 1+α and the asymptote (120) holds for s > 1+α.

Figure 5 :
Figure 5: A general rooted planar map with non-simple boundary (left) can be obtained uniquely by gluing rooted planar maps to each boundary vertex of a planar map with simple boundary (right).

Figure 6 :
Figure 6: An example of a loop-decorated planar map with a boundary (left) and its gasket (right).
(left).Each such loop-decorated planar map comes with a weight n #loops (2z * ) −#loop-decorated triangles non-loop faces f qdeg(f) , 95) for z > z * , where b = arccos(n/2)/π ∈ [0, 1] and C 1 and C 2 > 0 are constants.In particular W (z)| sing.∼ (z − z * ) m−b when z → z * .One should notice that this is precisely the scaling of the generalized Kazakov disk amplitude W (z) at the critical value g = g * when s = m − b + 1, since at g = g * we have explicitly