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Appendix B, we discuss the transition temperatures of the moments of the partition function in the disordered case.
II. MODEL AND OBSERVABLES
A. Definition of the model
−u1
la −u2
−u3 ld
−u4
FIG. 1: In this paper, we consider a Cayley tree of branching ratio K and a number L of generations (K = 2 and L =4 on the Figure), and we study the statistical physics of a directed polymer starting at the root in the presence of random energies (−u1, −u2, .., −uL) along the left-most bonds. A configuration of the directed polymer is characterized by an attached length 0 ≤ la ≤ L along these left-most bonds and a detached length ld = L − la (la = 1 and ld = 3 on the Figure). The delocalized phase corresponds to a finite la as L → +∞, whereas the localized phase corresponds to a finite ld as L → +∞. The order parameter is the contact density θ = la/L.
We consider a directed polymer of length L on a Cayley tree of branching ratio K. The total number of directed walks is simply
L YL = K (1)
The corresponding delocalized free-energy reads
F deloc T ln Y = TL ln K (2) L ≡− L − and corresponds to an entropy of (ln K) per bond. We now consider the random wetting model, where the left-most bonds are characterized by independent random energies ( ui) with i =1, 2, .., L (see Figure 1). As an example, they can be drawn from the Gaussian distribution of − 2 average value u0 > 0 and variance ∆
− 2 1 (ui u0) p(u)= e− 2∆2 (3) √2π∆2
B. Recurrence satisfied by the partition function
To write a recurrence for the partition function, it is convenient to label the random energies from the bottom, instead of the labelling from the root shown on Fig. 1
u˜i = uL i (4) −
The partition function ZL then satisfies the simple recurrence
Z = eβu˜L+1 Z + (K 1)Y (5) L+1 L − L 3 with the initial condition Z0 = 1. It is convenient to introduce the ratio with respect to the free partition function of Eq. 1
ZL ZL RL = L (6) ≡ YL K The recurrence then becomes
eβu˜L+1 (K 1) R = R + − (7) L+1 K L K with the initial condition R0 = 1.
C. Free-energy difference with the delocalized state
The difference between the free-energy F = T ln Z and the delocalized free-energy of Eq. 2 is given by L − L F diff F F deloc = T ln Z + T ln Y = T ln R (8) L ≡ L − L − L L − L The excess free-energy per monomer due to the wall in the thermodynamic limit L + → ∞ F F deloc ln R f lim L − L = T lim L (9) ∞ L + L L + L ≡ → ∞ − → ∞ characterizes the wetting transition : it vanishes in the delocalized phase f (T >Tc) = 0 and remains finite in the ∞ localized phase f (T D. Probability distributions of the attached length la and of the detached length ld Since there is no loop on the Cayley tree, the directed polymer cannot return to the wall after leaving it. A configuration can be thus decomposed into a length l attached to the wall starting at the root and a length l = L l a d − a detached from the wall (see Fig. 1). The thermal probability QL(la) to have exactly la links attached to the wall in a given disordered sample reads la β ui i=1 L la 1 e X δla,L + (K 1)K − − θ(la The numerator is the contribution to the partition function of the configurations that leave the wall after exactly la steps. The denominator is the partition function la L β ui L la 1 Z = e i=1 δ + (K 1)K − − θ(l L QL(la)=1 (12) lXa=0 It will be convenient to introduce also the probability distribution of the detached length l = L l d − a L l − d β ui i=1 ld 1 e X δld,0 + (K 1)K − θ(ld 1) PL(ld) QL(la = L ld)= − ≥ (13) ≡ − ZL 4 E. Order parameter : contact density A convenient order parameter of the transition is the contact density, which is directly related to the attached length la introduced in Eq. 10 l θ a (14) L ≡ L In particular, its thermal average in a given disordered sample is determined by the first moment of the distribution QL(la) of Eq. 10 L F. Internal Energy The internal energy E = ∂ ln Z = ∂ ln R (16) L − β L − β L satisfies the recursion (see Eq. 7 for the corresponding recursion for the reduced partition function RL) βu˜L+1 e RL( u˜L+1 + EL) EL+1 = − (17) eβu˜L+1 R + (K 1) L − and the initial condition E0 = 0. In terms of the attached length la introduced above (see Eq. 10), it can also be written as la L la β ui L la 1 u e i=1 δ + (K 1)K − − θ(l e = ∂β(βf ) (20) ∞ ∞ III. STATISTICAL PROPERTIES OF THE FREE-ENERGY In this section, we discuss the statistics of the free-energy difference of Eq. 8 which depends only on the variable RL introduced in 6. We first explain that RL is a Kesten random variable and describe its statistical properties. A. Analysis of the variable RL as a Kesten random variable The recursion of Eq. 7 takes the form of the recurrence of Kesten variable [14] RL+1 = aL+1RL + b (21) 5 where b = (K 1)/K is a constant and where the − eβui a = (22) i K are independent identically distributed random variables. The specific structure of a Kesten variable RL consists in a sum of products of random variables L L L R = a + b 1+ a (23) L i i i=1 j=2 i=j Y X Y This discrete form has for continuous analog the exponential functional L L dyF (y) = dxeRx (24) RL Z0 where F (x) is the random process corresponding to the random variables (ln ai) in the continuous limit. This{ type of} random variables, either in the discrete or continuous forms, appears in a variety of disordered systems, in particular in random walks in random media [15, 16, 17, 18, 19, 20, 21, 22, 24, 25], in the classical random field Ising chain [26, 27], and in the quantum random transverse field Ising chain [28, 29, 30]. Many properties have been thus already studied in great detail in these previous works. In the following, we cite and translate these known results for the present context. To describe the universal results near the critical point for large samples, there are only two relevant parameters : (i) the first important parameter is the averaged value F ln a = βu ln K (25) 0 ≡ i − The critical point corresponds to the vanishing condition ln ai = 0, so that the critical temperature of the present random wetting model reads u T = (26) c ln K (In particular, for the Gaussian distribution of Eq. 3, the critical temperature is Tc = u0/(ln K) and is independent of the variance ∆2.) (ii) the second important parameter is the variance of (ln ai) that will be denoted by 2σ (ln a )2 (ln a )2 = β u2 (u )2 (27) ≡ i − i i − i N to keep the same notations as in Refs. [22, 25, 30]. In particular, from the Central Theorem, the sum ΣN = (ln ai) i=1 is then distributed asymptotically for large N with the Gaussian distribution X − 2 1 (ΣN NF0) PN (ΣN ) e− 4σN (28) N ≃ √ →∞ 4πσN B. Delocalized phase T >Tc For T >Tc, the parameter of Eq. 25 is negative F0 < 0 and the random variable RL remains a finite random variable as L . Its probability distribution P (R ) is known to display the power-law decay [14, 26, 27] → ∞ ∞ ∞ 1 P (R ) 1+µ(T ) (29) ∞ ∞ R∞∝ →∞ R ∞ where the exponent µ(T ) > 0 is determined by the condition eµβui 1= aµ = (30) i Kµ 6 deloc In conclusion, the difference of Eq. 8 between the free-energy FL and the delocalized free-energy FL remains a finite random variable as L → ∞ diff deloc diff F FL FL F T ln R (31) L L ∞ ≡ − →∞≃ ≡− and its distribution presents the following exponential tail for F (see Eq. 29) diff → −∞ diff βµ(T )Fdiff Pdeloc(F ) e (32) Fdiff≃ →−∞ Example : for the case of the Gaussian distribution of Eq. 3, the condition of Eq. 30 becomes 2 2 2 µβu + µ β ∆ e 0 2 1= (33) Kµ and the exponent µ reads 2T (T ln K u ) 2(ln K)T (T T ) µGauss(T )= − L = − c (34) ∆2 ∆2 C. Critical point T = Tc Exactly at criticality T = Tc, the parameter of Eq. 25 vanishes F0 = 0. It is then known (see for instance [22, 23, 30] and references therein) that the leading behavior is of order 1/2 ln RL(Tc) 2(σL) wc (35) L ≃ →∞ where wc is a positive random variable of order O(1). Moreover, it can be shown (see [30] and references therein) that the distribution of wc is the half-Gaussian distribution (cf Eqs (24) and (32) in [30]) 2 w2 P (w ) θ(w 0) e− (36) c ≃ c ≥ √π D. Localized phase T For T ln RL(T 1 w2 P (w) e− (38) ≃ √π E. Summary of the critical behavior of the free-energy and energy deloc In conclusion, the difference of Eq. 8 between the free-energy FL and the delocalized free-energy FL displays the following critical behavior diff FL (T >Tc) = Fdiff of order O(1) F diff (T ) = T 2(σL)1/2w with w > 0 of order O(1) L c − c c c F diff (T e (T >Tc) =0 (42) ∞ e (T IV. STATISTICAL PROPERTIES OF THE ATTACHED AND DETACHED LENGTHS In this section, we discuss the statistical properties of the attached length la or of the complementary detached length ld whose distributions have been introduced in Eq. 10 and Eq. 13 respectively. This allows to analyse also the statistics of the contact density of Eq. 14, which represents the order parameter and of the internal energy of Eq. 18. A. Typical behavior of QL(la) in the delocalized phase In the delocalized phase, the attached length l will remain a finite random variable as L . The typical decay a → ∞ of QL(la) is then given by (Eq. 10) K 1 ln Q (l ) = βl u + (L l ) ln K + ln − ln Z (44) L a a − a K − L K 1 l (ln K βu) + ln − + (L ln K ln Z ) (45) ≃− a − K − L i.e. the decay is governed by the typical correlation length 1 = ln K βu (46) ξtyp − that diverges with the correlation length exponent νtyp =1 (47) as in the pure case ( νpur = 1 of Eq. A8). B. Typical decay of PL(ld) in the localized phase In the localized phase, the detached length l will remain a finite random variable as L + . The typical decay d → ∞ of PL(ld) is then given by K 1 ln P (l ) = β(L l )u + l ln K + ln − ln Z (48) L d − d d K − L K 1 l (βu ln K) + ln − + (Lβu ln Z ) (49) ≃− d − K − L i.e. the decay is governed by the typical correlation length 1 = βu ln K (50) ξtyp − that diverges with the correlation length exponent νtyp =1 (51) as in the pure case ( νpur = 1 of Eq. A17). 8 C. Statistics over the samples of the probability qL = QL(la = 0) of zero contacts To understand why typical and disordered averaged behaviors can be different for the distributions QL(la) and PL(ld), it is convenient to consider first the particular case of the probability of zero contacts L 1 (K 1)K − (K 1) 1 qL QL(la =0)= PL(ld = L)= − = − (52) ≡ ZL K RL because it depends only on the Kesten variable RL whose statistics has been discussed in detail in section III. We also note that qL coincides with the expression of the flux JL in the Sinai model, and we refer to [22] where many exact results have been derived for its probability distribution (see in particular Eq. (6.10) in [22]). In the following, we concentrate on the critical region. From the statistics of RL described in section III, one has in particular that (i) in the delocalized phase, q remains a finite random variable as L + L → ∞ (ii) in the localized phase, the probability qL becomes exponentially small in L wc2√σL (ii) exactly at criticality, the variable qL behaves as qL e− where wc is a positive random variable of order O(1) distributed with the half-Gaussian distribution of Eq.∼ 36. As discussed in [30], an important consequence of the critical statistics of Eq.36 is that the averaged value is dominated by the rare samples having an anomalously small value of the scaling variable 0 w 1/√L, and since the probability distribution P (wc) is finite at wc = 0, one obtains the slow power-law decay≤ ≤ 1 qL(Tc) (53) L ∝ √ →∞ L whereas the typical behavior is given by the exponential decay ln qL(Tc) √L (54) L →∞∝ − We refer to [30] and references therein for the discussion of similar examples of averaged values dominated by rare events in the field of quantum spin chains. D. Notion of “ Infinite Disorder Fixed Point” The probability distribution QL(la) of the attached length la in a given sample has the form of a Boltzmann measure e U(la) Q (l )= − (55) L a L U(la) e− lXa=0 over the random walk potential la la U(l )= β u + l ln K = ln a (56) a − i a − i i=1 i=1 X X The continuum analog of Eq. 55 is then the Boltzmann measure over a Brownian potential (see Eq. 10 in [25]). We refer to [25] and references therein for more detailed discussions, but the essential property is that the Boltzmann measure over a Brownian valley is very concentrated (in a region of order O(1)) around the absolute minimum of the Brownian potential. Within our present notations, this means that in a given disordered sample at a given temperature, the thermal probability distribution QL(la) is concentrated in a finite region around the point la∗ where the random potential of Eq. 56 reaches its minimum on the interval 0 l L ≤ a ≤ Q (l ) δ(l l∗) (57) L a ≃ a − a This means that in a given sample, the contact density of Eq. 14 is essentially given by its thermal average l θ <θ >= a∗ (58) L ≃ L L 9 E. Sample-to-sample fluctuations of the contact density We now discuss the sample-to-sample fluctuations of la∗ and of the contact density. The disorder average of the probability distribution QL(la) can be obtained as Q (l ) δ(l l )= π (l ) (59) L a ≃ a − a∗ L a where πL(la∗) is the probability distribution of the minimum la∗ over the samples. Taking into account the two parameters F0 (Eq. 25) and σ (Eq. 27) that characterize the large scale properties of the random walk U(la), the probability distribution πL(la∗) of the minimum la∗ can be obtained as (F0) ( F0) ψ (L la∗)ψ − (la∗) πL(la∗)= L − (60) dlψ(F0)(L l )ψ( F0)(l ) 0 − a∗ − a∗ where R + ∞ ψ(F0)(l)= dUG(F0)(U,l) (61) Z0 and where G(F0)(U,l) represents the probability for a biased random walk to go from (0, 0) to (U,l) in the presence of an absorbing wall at U =0− (see for instance [30] for very similar calculations) − 2 (F ) U (U F0l) G 0 (U,l)= e− 4σl (62) 2√π(σl)3/2 The critical behavior can be analyzed as follows. 1. Sample-to-sample fluctuations at criticality At criticality where F0 = 0, the function ψ of Eq. 61 reads + 2 F =0 ∞ U U 1 ψ 0 (l)= dU e− 4σl = (63) 2√π(σl)3/2 √πσl Z0 The distribution of Eq. 60 for the position la∗ then reads (Tc) 1 QL(la∗)= π (la∗)= (64) L π l (L l ) a∗ − a∗ p As a consequence, we find that the contact density θ la∗/L is distributed at criticality over the samples with the law ≃ 1 T (θ)= (65) P c π θ(1 θ) − p 2. Sample-to-sample fluctuations in the delocalized phase In the delocalized phase where F0 < 0, the position la∗ of the minimum on [0,L] of the Brownian potential of positive drift remains finite as L + , and the probability distribution of Eq. 60 can be obtained as → ∞ ( F0) (T >Tc) ψ − (la∗) πL (la∗) L + (66) ≃ →∞ ∞ ( F0) 1 dlψ − (l) with R + 2 2 1/2 + 2 (U+F0la) F0 2 ( F ) ∞ U F0 (σla) ∞ la (1+z) ψ − 0 (l )= dU e− 4σla = dzze− 4σ (67) a 2√π(σl )3/2 σ22√π Z0 a Z0 10 In particular, it decays exponentially as F 2 (T >Tc) 0 4σ la QL(la)= πL (la) e− (68) la ∝+ → ∞ with the correlation length 4σ 1 ξa = (69) 2 + 2 F0 T ∝Tc (T Tc) → − This should be compared with the typical decay of Eq. 45. As a consequence, the finite-size scaling properties in the critical region involves the correlation length exponent νFS =2 (70) and not the typical correlation exponent that appears in Eq. 47. 3. Sample-to-sample fluctuations in the localized phase In the localized phase, where F > 0, it is the detached length l∗ = L l∗ that remains finite as L + , and the 0 d − a → ∞ disorder-averaged probability of the detached length ld can be obtained as (F0) (T 2 F0 ld PL(ld) e− 4σ (73) ld ∝+ → ∞ with the correlation length 4σ 1 ξ = (74) d 2 − 2 F0 T ∝Tc (Tc T ) → − This should be compared with the typical decay of Eq. 49. Again, this means that the finite-size scaling properties in the critical region are governed by the correlation length exponent νFS =2 of Eq. 70. V. DISCUSSION A. Lack of self-averaging of the order parameter at criticality Outside criticality, the contact density is self-averaging in the thermodynamic limit L + → ∞ (θ) = δ(θ 1) (75) PT B. Physical interpretation of two correlation length exponents The presence of two different correlation length exponents νav = 2 and νtyp = 1 whenever the disorder is equivalent to a Brownian potential U(x) is well known in other contexts, in particular for random walks in random media [19, 25], for the quantum transverse field Ising chain [4, 30] and more generally in other models described by the same “Infinite Disorder Fixed Point” (see the review [6] and references therein). The physical interpretation is the following [4] : (i) the first length scale corresponds to the length ξtyp where the mean value [U(L) U(0)] = F0L is of order one, which yields − 1 ξtyp νtyp with νtyp =1 (78) ∼ F0 (ii) the second length scale corresponds to the length ξ where most of the samples indeed have (U(L) U(0)) av − ∼ F0L √σL of the same sign of the mean value, i.e. the scale √σL of the fluctuations should be of the same order of the mean± value, which yields 1 ξav ν with νav =2 (79) ∼ F0 We have found that the finite-size scaling properties for disorder-averaged values over the samples are governed by ν = ν = 2 which actually saturates the general bound ν 2/d [34], where d is the dimensionality of the FS av FS ≥ dis dis disorder (here ddis = 1), whereas the typical exponent is smaller νtyp = 1. This possibility of a first order transition that remains first order in the presence of quenched disorder has been already discussed in [34] and in Sec. VII A of Ref. [4]. VI. CONCLUSIONS AND PERSPECTIVES In this paper, we have studied the random wetting transition on the Cayley tree. We have obtained that the transition which is first order in the pure case, remains first-order in the presence of disorder, but that there exists two diverging length scales in the critical region : the typical correlation length diverges with the exponent νtyp = 1, whereas the averaged correlation length diverges with the bigger exponent νav = 2 which governs the finite-size scaling properties over the samples. We have describe the relations with the previously studied models that are governed by the same “infinite disorder fixed point”. We have given detailed results on the statistics of the free-energy, and on the statistics of the contact density θ = la/L which constitutes the order parameter in wetting transitions. In particular, we have obtained that at criticality, the contact density is not self-averaging but remains distributed over the samples in the thermodynamic limit, with the distribution Tc (θ)=1/(π θ(1 θ)). As explained at the beginning, our physical motivationP to consider such− a model of random wetting on the Cayley tree was to better understand the similarities and differences withp two other types of models involving directed polymers and frozen disorder described in the points (i) and (ii) of the Introduction. The results given in the present paper suggest the following conclusions : (i) The freezing transition of the directed polymer in a random medium on the Cayley tree [7] is of a very different nature, since it is the exponent ν = 2 which governs the free-energy singular part (instead of νtyp = 1 here). The appearance of a smaller exponent ν′ = 1 in some finite-size properties in the critical region [8, 9] should be then explained with another mechanism. (ii) Let us now compare with the wetting and Poland-Scheraga model of DNA denaturation with a loop exponent c> 2 that we have studied numerically in [12, 13]. We find that the random wetting model considered in the present paper ( that corresponds to the limit of loop exponent c of the model studied in Ref. [12] ) presents very similar → ∞ critical behavior : in both cases, the transition remains first order with νtyp = 1 and a finite contact density, but the finite-size scaling properties over the samples involve the exponent νFS = 2. We thus hope that the exact results obtained here for the special case c will help to better characterize the transition in the region of finite-loop exponent 2 APPENDIX A: PROPERTIES OF PURE WETTING TRANSITION ui = u0 In this Appendix, we describe the critical properties of the pure wetting transition on the Cayley tree, to compare with the disordered case considered in the text. 12 1. Finite-size partition function When all the energies ui take the same value u0, the ratio RL of Eq. 6 simply reads L 2 L 1 eβu0 (K 1) eβu0 eβu0 eβu0 − Rpure = + − 1+ + + + L K K K K · · · K " # βu L L e 0 eβu0 (K 1) K 1 = + − (A1) − βu K K × e 0 1 K − The critical temperature Tc then corresponds to the point where the energy per link u0 exactly compensates the entropy per link (ln K) of the free walk u0 eβcu0 = K T pure = (A2) → c ln K 2. Delocalized phase eβu0 < K In the delocalized phase, the leading term of Eq. A1 remains finite pure (K 1) RL (T >Tc) − + ... (A3) L ≃ (K eβu0 ) →∞ − and the excess free-energy per monomer due to the wall in the thermodynamic limit L + (Eq. 9) vanishes → ∞ f (T >Tc)=0 (A4) ∞ The finite-size behavior of the full free-energy difference of Eq. 8 reads near criticality diff deloc (K 1) 1 FL (T >Tc)= FL(T >Tc) FL = T ln RL = T ln −βu Tc ln (A5) − − − (K e 0 ) T ≃T + − T Tc − → c − In the delocalized phase, the probability distribution QL(la) of the attached length la of Eq. 10 remains finite as L → ∞ βu0 la e βu0 QL(la) PL(ld = L la) 1 Ke− − = Q (la) (A6) ≡ − L ≃ − K ∞ →∞ It decays exponentially la ξ (T ) Q (la) e− deloc (A7) ∞ la ≃+ → ∞ where the correlation length reads pure 1 1 ξdeloc(T >Tc)= ν with νpure = 1 (A8) u0(βc β) T ∝T + (T Tc) pure − → c − The thermally averaged contact density <θL > of Eq. 15 vanishes as 1 ∞ 1 <θL > laQ (la)= (A9) L L ∞ L(Ke βu0 1) →∞≃ − lXa=0 − i.e. in the critical region, it behaves as 1 <θL > = (A10) L ≃ L(β β) →∞ c − 13 3. Localized phase eβu0 > K pure In the localized phase T L βu0 pure e (K 1) RL (T βu0 (K 1)u0e EL(T e (T ld K 1 eβu0 − δ + θ(l 1) − ld=0 d ≥ K K P (ld)= (A15) ∞ (K 1) 1+ βu− K e 0 1 " “ K − ” # It decays exponentially ld ξ T P (ld) e− ( ) (A16) ∞ ld ≃+ → ∞ where the correlation length diverges as 1 1 ξpure(T 4. Critical point eβcu0 = K Exactly at criticality, the ratio of Eq. A1 reads (K 1) Rpure(T )=1+ − L (A18) L c K so that the free-energy difference is logarithmic in L diff pure deloc (K 1) F (Tc)= F (Tc) FL (Tc)= Tc ln RL = Tc ln 1+ − L Tc ln L (A19) L L − − − K L ≃ − →∞ The recurrence of Eq. 17 for the energy becomes KR K + (K 1)L E = L ( u + E )= − (u + E ) (A20) L+1 KR + (K 1) − 0 L K + (K 1)(L + 1) 0 L L − − leading to L(L 1) 1+ 2K pure KL + (K 1) − (L 1) (K 1)(L 1) 2 − − EL (Tc)= u0 − = u0 − K (A21) − K + (K 1)L − 2 × 1+ (K 1)L − − 14 so that the energy per monomer in the thermodynamic limit L is → ∞ pure u0 e (Tc)= (A22) ∞ − 2 pure pure i.e. exactly in the middle of the jump between e (T (K 1) − δla,L + K θ(la APPENDIX B: DISORDERED CASE : TRANSITION TEMPERATURES Tn FOR THE AVERAGED n MOMENTS ZL OF THE PARTITION FUNCTION n Since in other disordered models, one sometimes consider the series Tn of critical temperatures for the moments ZL of the partition function, we discuss their properties in the present Appendix. 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