Directed polymers on a disordered tree with a defect subtree
Neal Madras Gökhan Yıldırım
math.PR math-ph math.MP


]Directed polymers on a disordered tree with a defect subtree


Department of Mathematics and Statistics, York University
 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada



madras@mathstat.yorku.ca


Department of Mathematics, Bilkent University
06800 Ankara, Turkey
gokhan.yildirim@bilkent.edu.tr


We study the question of how the competition between bulk disorder and a localized microscopic defect affects the macroscopic behavior of a system in the directed polymer context at the free energy level. We consider the directed polymer model on a disordered d-ary tree and 
represent the localized microscopic defect by modifying the disorder distribution at each vertex in a
single path (branch), or  in a subtree, of the tree.
The polymer must choose between
following the microscopic defect and 
finding the best branches through the bulk disorder. We 
describe 
three possible phases, called the fully pinned, partially pinned and depinned phases. 
When the microscopic defect is 
associated only with a single branch, we compute the free energy and the critical curve of the model, 
and show that the partially pinned phase 
does not occur.
When the localized microscopic defect is
associated with a non-disordered 
regular 
subtree of the disordered tree, the picture is more complicated. 
We prove that all three phases are non-empty below a critical temperature, and that the partially 
pinned phase disappears above the critical temperature. 





82B44  82D60 60K35 


Keywords: Directed polymers, free energy, bulk disorder, microscopic defect





[
    Gökhan Yıldırım
    December 30, 2023
=====================





§ INTRODUCTION


Directed polymers in a random environment are typical examples of models used to study the behavior of a one-dimensional object interacting with a disordered environment. In the mathematical formulation of these models, paths of a directed walk on a regular lattice or tree represent the directed polymer while an independent and identically distributed (i.i.d.) collection of random variables attached to the vertices
of the lattice/tree correspond to the random environment (bulk disorder). 
Each path is assigned a Gibbs weight corresponding to the sum of the random variables of the visited vertices.
The polymer's interaction with the random environment is controlled by a parameter, β, which represents the inverse temperature. The main questions are whether there exist different phases in the model depending on the temperature which manifest the effect of the disorder on the large scale behavior (diffusive versus superdiffusive) of the polymer, and how the phases can be characterized <cit.>. 
The earliest example of the model studied in the physics literature <cit.> 
(and then rigorously in <cit.>) was a 1+1 dimensional lattice case as a model for the interface in 
a two-dimensional Ising model with random exchange interaction. Since then it has been used in models of various growth phenomena: formation of magnetic domains in spin-glasses <cit.>, vortex lines in superconductors <cit.>, roughness of crack interfaces <cit.>, and the KPZ equation <cit.>. The last twenty years have witnessed many significant results on the problems related to directed polymer models and more general 
polymer models. For a comprehensive introduction and an up-to-date summary of the results and methods for both the lattice and tree version of the directed polymer model, see the lecture notes <cit.>. For more general polymer models, see <cit.>. 

A different direction of research considers polymers in a deterministic 
environment with a localized microscopic defect.  
A primary example is the case of an interfacial layer between two fluids, 
modelled by a plane in a 3-dimensional lattice (called a “defect plane” in some contexts),
such that each monomer of a polymer is energetically rewarded if it lies in this layer.
A related model is the situation that the monomers are attracted to an impenetrable 
wall of a container;
in this case, the polymer lives in a half-space bounded by an attracting plane.
There is typically a critical value of β above which a positive fraction 
of the polymer is pinned or adsorbed to the surface, and below 
which the polymer is mostly free of the surface—that is, it is depinned or desorbed.
It is generally expected that the critical β is strictly positive for an impenetrable 
boundary, and equal to zero for a penetrable boundary.  This problem can be solved 
exactly for directed polymers <cit.>.  For the self-avoiding
walk model of polymers, the impenetrable result has been proven <cit.>, while the 
penetrable case remains open, but can be proved under an extremely weak hypothesis
<cit.>. 
In the very special case of self-avoiding walks at an impenetrable boundary on the honeycomb lattice, 
the exact critical value has been determined in <cit.>. 
Pinning problems also arise elsewhere, notably the context of high-temperature 
superconductors <cit.>.



 §.§ The bulk disorder versus a localized microscopic defect.




In this paper, we shall study the question of how the competition between bulk disorder and a localized microscopic defect affects the macroscopic behavior of a system as reflected in pinning phenomena of directed polymers.
In the directed polymer on a disordered tree model,  we add a fixed potential u to each vertex 
on a branch or a subtree of the tree which represents the localized microscopic defect. 
Roughly speaking, the polymer must choose between following the localized microscopic defect and finding the best branch(es) through the bulk disorder. We see that there are three possible phases depending on the defect structure (a single branch versus a subtree) and the model parameters (β, u): 


  - Fully pinned phase, R_FP: the partition function is dominated by polymer configurations that spend almost all their time in the defect structure. 

  - Depinned phase, R_D: the partition function is dominated by polymer configurations
that spend hardly any time in the defect structure.

  - Partially pinned phase, R_PP: the partition function is dominated by polymer configurations that spend a positive fraction (but not close to all) of their time inside the defect structure.


For the formal definition of each phase, see Definition <ref> in section <ref>.


In the (nonrigorous) physics literature, this problem has been studied extensively in the lattice version of the directed polymer model <cit.> but there have been disagreeing predictions for the 1+1 dimensional lattice version as to whether the polymer follows the defect line as soon as the potential level u is greater than 0; for more details see section <ref>. 
For some rigorous partial results in this direction, see <cit.>. We consider this problem in the tree version of the directed polymer model which can be viewed as a mean field approximation of the lattice case.





In order to study our problem precisely, we first present some definitions and introduce some notation related to the directed polymers on disordered trees, a model introduced in <cit.>. Let  be a rooted d-ary tree, in which every node  has exactly d offspring (d≥ 2). We label the nodes of  by two integers (k,j) where k corresponds to the generation and j∈{1,2,⋯,d^k} numbers the nodes within the k^th generation from left to right. The root is labeled as =(0,1). See Figure <ref>. An infinite directed path from the root is called a branch of the tree.

We assume that every node x=(k,j) of the tree  has an associated random variable denoted by V(x) or V_k,j that represents the random disorder at that node, all independent. 

The Hamiltonian of the model is defined as

    V⟨ W⟩ := ∑_y∈ W∖{} V(y)

where W is a directed path in  from the root  to some node in the n^th generation. 

In the homogeneous disorder (HD) model,
all the random variables V(x) have the same distribution as some 
non-degenerate random variable V with 

    λ(β):=log𝐄[e^β V] <∞ for all  β∈ℝ.


The partition function of the HD model is defined as

    Z_n^HD(β)    := ∑_W:→ (n,·)   e^β V⟨ W⟩

where the sum is over all directed paths W in  from the root  to some node in the 
n^th generation. The parameter β represents the inverse temperature.



The free energy of the HD model is defined to be 

    ϕ(β):=lim_n→∞1/nlog Z^HD_n(β).




For each β, this limit exists and is constant almost surely.   It is computed explicitly in <cit.> as

    ϕ(β)={[         λ(β) +log d         if   β <β_c; β/β_c(λ(β_c)+log d)          if   β≥β_c ].

where 
the critical inverse temperature β_c is the positive root of 
λ(β)+log d  = βλ'(β), or is +∞ if there is no root; 
see Lemma <ref> in section <ref> for details.

The defect structure is incorporated into the model by assigning random variables from a different distribution
to the vertices in a part of the tree. 
Let  be the “left-most” d_1-regular subtree of the d-regular tree , with the same 
 root  (for the precise definition see the beginning of section <ref>). 
 We assume that there are two possible distributions for V(x), 
which we shall call V and Ṽ:

  If x∈, then V(x) has distribution Ṽ.   

  If x∈∖, then V(x) has distribution V.

We assume that V satisfies Equation (<ref>).  We shall consider two special cases:

  Case I (Shift defect): There is a real constant u such that the   distribution of Ṽ is  V+u. 
    

  Case II (Nonrandom defect): There is a real constant u such 
  that 𝐏(Ṽ=u) = 1. 





  §.§.§ Polymers on non-disordered trees with a defect subtree.


As a first case, we shall consider a directed polymer model on a deterministic d-regular tree , no bulk disorder, and identify the localized microscopic defect with a d_1-regular subtree  of  by 
placing a fixed potential u at each vertex of  and potential 0 elsewhere in . 
That is, we have 𝐏(V=0) = 1 and 𝐏(Ṽ=u) = 1.  


We define the free  energy as

    f^Det(β,u):= lim_n→∞1/nlog Z_n^Det(β,u)

where Z^Det_n(β,u) is the partition function of the model. Note that f^Det(β,0)  = log d.

The critical curve is defined as

    u^Det_c(β):=inf{u∈ℝ: f^Det(β,u)>log d}.

The following result is straightforward to prove (see Section <ref>):

 For any β≥ 0 and u∈ℝ, we have

    f^Det(β,u) = max{β u+log d_1,log d}

and hence 

    u^Det_c(β)  = log (d/d_1)/β .


We interpret the critical curve as follows.  When u<u_c(β), then f^Det(β,u) =log d,
which shows that the free energy is dominated by walks that are entirely (except for the root)
outside of ; there are (d-d_1)d^n-1 such walks, each with weight 1 in the 
partition function.  This corresponds to the desorbed or depinned phase. 
In contrast, when u>u_c(β), then f^Det(β,u) =β u+log d_1,
which shows that the free energy is dominated by walks that are entirely 
in ; there are d_1^n such walks, each with weight e^β nu.
This corresponds to the fully adsorbed or fully pinned phase.   




  §.§.§ Polymers on disordered trees with a defect branch.



Next, we shall consider a directed polymer model on a 
d-regular tree  with bulk disorder and a one-dimensional microscopic shift defect.
Specifically, we identify the defect with the leftmost branch  of the tree 
by adding a fixed potential u to each vertex of ; 
that is, the distribution of Ṽ is  V+u.
See Figure <ref>. 
Therefore, for a directed path W from the root to some node in the n^th generation, 
the Hamiltonian is 

    V⟨ W⟩ := ∑_y∈ ∩  (W∖{}) (V(y)+u)  +
    ∑_y∈ (∖) ∩ (W∖{}) V(y)  .



Then the free energy of the model is defined as 

    f^Br(β,u):=lim_n→∞1/nlog Z^Br_n(β,u)

where Z^Br_n(β,u) is the partition function of the model, defined as in the right-hand side of
Equation (<ref>).
The existence of the limit in Equation (<ref>) is part of the assertion of 
Theorem <ref> below.   Observe that f^Br(β,0) = ϕ(β) (recall Equation (<ref>)).

We define the critical curve as

    u^Br_c(β):=inf{u∈ℝ: f^Br(β,u)>ϕ(β)}.

In our next result, we compute the free energy and the critical curve explicitly. In the statement of the theorem, the quantity β_c is the critical inverse temperature for the 
homogeneous disorder model, see Equation (<ref>) and section <ref>.

 For any β≥ 0 and u∈ℝ, we have almost surely

    f^Br(β,u)=max{β u+βμ,ϕ(β)}

where μ=𝐄(V).
Hence

    u^Br_c(β)={[    1/β(λ(β) +log d)-μ          if   β < β_c; 1/β_c(λ(β_c)+log d)-μ         if   β≥β_c  . ].


We also see that for this model the partially pinned phase,  R_PP, is always empty.   Indeed, we show in the proof of Theorem <ref> (see Section 
<ref>) that β u+βμ is the contribution to the free energy from the path W
that lies in .
 
Note that for a non-degenerate random variable V,  e^λ(β)=𝐄(e^β V)>e^β𝐄(V)=e^βμ. Therefore

    u^Br_c(β)>{[      log d/β if   β < β_c;    log d/β_c  if   β≥β_c. ].

From Equations (<ref>) (with d_1=1) and (<ref>), we  see that quenched 
randomness shifts the critical curve, that is, u^Br_c(β)>u^Det_c(β) for all β>0. 




  §.§.§ Polymers on disordered trees with a non-disordered defect subtree.


 In this section, we shall consider a different microscopic defect structure that is identified with a deterministic d_1-regular subtree  of the d-regular tree , and we identify the bulk disorder with the vertices in ∖; that is, there is a real constant u such that 
 Ṽ≡ u.  See Figure <ref>.
Therefore for a directed path W from the root to some node in the n^th generation, 
the Hamiltonian is 

    V⟨ W⟩ := ∑_y∈∩ (W∖{}) u     +
    ∑_y∈ (∖)∩ (W∖{}) V(y)  .

We denote the partition function of the model with a defect subtree by Z^ST_n(β,u). 
For this model, it is not obvious how to prove that the limiting free energy exists almost surely; see the beginning of section <ref>.





We shall define the following functions:  
 
    F(β)       :=    1/β (λ(β)+log d -log d_1)   
    
          
          J(β)     :=    1/β( ϕ(β)-log d_1-
          [λ(β)+log d-ϕ(β)]log d_1/λ(2β)-2λ(β)-log d) .


Our main result for this model is the following.   

The proof appears in section <ref>. See also Figure <ref>. 
 

     
(a)  For every β∈ [0,β_c],  we have

    (β,u)     ∈ {[             R_FP    if    u≥ F(β);              R_D if    u≤ F(β)  . ].

(b)
For every β>β_c, we have

    (β,u)     ∈ {[                  R_FP        if     u≥ F(β);                   R_D       if    u≤ F(β_c);                  R_PP     if    J(β)<u<F(β);             R_D∪ R_PP if    F(β_c)<u≤ J(β). ].


We also prove in Proposition <ref> that F(β_c)<J(β)<F(β) whenever β>β_c.
This shows that for every β>β_c, there is a value of u such that (β,u)∈ R_PP.
That is, the partially pinned phase appears as soon as β exceeds β_c.



In a different direction of research <cit.>, Basu, Sidoravicious and Sly considered the question of “how a localized microscopic defect, even if it is small with
respect to certain dynamic parameters, affects the macroscopic behavior of a system” in the context of two classical exactly solvable models: Poissonized version of Ulam's problem of the maximal increasing
sequence and the totally asymmetric simple exclusion process. In the first model, by using a Poissonized version of directed last passage percolation on ℝ^2, they introduced the microscopic defect by adding a small positive density of extra points along the diagonal line. In the second, they introduced the microscopic defect by slightly decreasing the jump rate of each particle when it crosses the origin. They showed that in Ulam's problem the time constant increases, and for the exclusion process the flux of particles decreases. Thereby, they proved that in both cases the presence of an arbitrarily small microscopic defect affects the macroscopic behavior of the system, and hence settled the longstanding “Slow Bond Problem” from statistical physics.


The rest of the paper is organized as follows.   In section <ref>, we introduce some notation, 
review the directed polymer on disordered tree model, and summarize the main existing results which we use in this paper. In section <ref>, we prove our results:  
Theorem <ref> is proved in section <ref>,
Theorem <ref> in section <ref>, 
and Theorem <ref> in section <ref>.
 We conclude by discussing our results and some related models in section <ref>.

For two random variables X and Y, we use the notation 
X d= Y to denote that they have the same distribution. If a probability statement is true 
with probability one, then we use the phrase “almost sure,” abbreviated “a.s.”.






 §.§ Polymers on trees with homogeneous disorder.


In this section, we present some definitions and review the main results related to directed polymers on disordered trees. Let  be a rooted d-ary tree, in which every node  has exactly d offspring (d≥ 2). Recall that we label the nodes of  by two integers (k,j) where k corresponds to the generation and j∈{1,2,⋯,d^k} numbers the nodes within the k^th generation. 
The root is  =(0,1).  The set of offspring of node (k,j) is 
{(k+1,(j-1)d+ℓ)  : 1≤ℓ≤ d}. See Figure <ref>. If x=(k,j), then we say that k is the generation or height of x, and we write
k= Height(x).
We assume that every node x=(k,j) of the tree  has an associated random variable denoted by V(x) or V_k,j that represents the random disorder at that node, all independent and with the same distribution as V.

Define

    f(β):=λ(β)+log d-βλ'(β)   for  β≥ 0

where λ comes from Equation (<ref>).

Note that λ is a strictly convex function of β, and therefore we have 
f'(β)<0 and f(β)<log d for all β>0.

For the proof of the following lemma, see <cit.>.


f has a unique positive root if and only if either


  - V is unbounded, or

  - w:=esssup V is finite and 𝐏(V=w)<1/d.

We use β_c to denote the unique positive root of f. If no solution exists, then β_c=∞.



Recall that Z^HD_n(β) denotes the homogeneous disorder partition function defined in Equation (<ref>).


The following positive martingale (M_n(β), ℱ_n)_n≥ 0 has
played a  crucial role in the analysis of the model:

    M_n(β):=Z^HD_n(β)/𝐄 Z^HD_n(β)

where ℱ_n=σ{V(x): Height(x)≤ n}
is the σ-algebra generated by all the random variables between generation 1 and n. The martingale methods are first used in <cit.> in the lattice version of the directed polymer model
 and then in <cit.> for the tree version. From the Martingale Convergence Theorem, it follows that M(β):=lim_n→∞M_n(β) exists almost surely and Kolmogorov's zero-one law implies that 𝐏(M(β)=0)∈{0,1} because {M(β)>0} is a tail event. 
 It is known that <cit.>
 
    M(β)>0       almost surely for all    0≤β <β_c
     
    M(β)=0       almost surely for all   β≥β_c
 
where β_c comes from Lemma <ref>.
The first case is called the weak disorder regime and the second case is called the strong disorder regime <cit.>. 
Recall that the critical inverse temperature β_c also marks a phase transition in the model at the level of the
free energy ϕ which is defined in Equation (<ref>). 


The strong disordered regime can be considered as the energy dominated or localized phase as a single polymer configuration supports the full free energy whereas the weak disorder regime can be considered as the entropy dominated or delocalized phase as the full free energy is supported by a random sub-tree of positive exponential growth rate, which is strictly smaller than the growth rate of the full tree <cit.>. Note also that M_n(β) converges to zero exponentially fast for β>β_c, but even though β_c is in the strong disorder regime the decay rate of M_n(β_c) is not exponential <cit.>. 

The following concentration result is proven in Proposition 2.5 of <cit.> for the partition function of the lattice version of the directed polymer model, and it is easy to see that it also holds true for the tree version of model.

For any ϵ>0 and β≥ 0, there exists N:=N(β,ϵ) such that

    𝐏(|log Z^HD_n(β) -𝐄log Z^HD_n(β)|≥ nϵ) ≤ exp(-ϵ^2/3n^1/3/4),      n≥ N  .


By combining Equations (<ref>) and (<ref>), we also get

    ϕ(β)=lim_n→∞1/n𝐄log Z^HD_n(β).


Observe that 

    𝐄Z^HD_n(β) =  d^n(e^λ(β))^n =  e^nλ(β)+nlog d

 and hence

    lim_n→∞1/nlog𝐄Z^HD_n(β)=λ(β) +log d.

We also note that 

    ϕ(β) ≤ λ(β)+log d   for every β

(for example, by Equations (<ref>) and (<ref>) and Jensen's inequality).  Indeed,
Equation (33) of <cit.> tells us that

    ϕ(β) <  λ(β)+log d   for every β>β_c.




§ PROOFS OF THE MAIN RESULTS


Before we  prove our results, we introduce some more notation. 
We assume that 1≤ d_1<d.
Let  be the “left-most” d_1-regular subtree of the d-regular tree , with the same 
 root , where “left-most” is interpreted as follows.
For a node x∈, let D̃(x) be the set of nodes in  whose parent is x,
and let D(x) be the set of nodes in ∖ whose parent is x. 
Using the notation x=(k,j), we specify

    D̃(x)       :=    { (k+1,d(j-1)+ℓ )  :   1≤ℓ≤ d_1}
    
         D(x)       :=    { (k+1,d(j-1)+ℓ)  :   d_1<ℓ≤ d} .

For d=3, the cases d_1=1 and d_1=2 are depicted in  Figures <ref> and <ref> 
respectively. 
Observe that

    |D̃(x)|  =  d_1   and |D(x)| =  d-d_1  
       for x∈.


Observe that for every directed path W=(w(0),w(1),…,w(n)) with w(0)= and 
Height(w(n))=n, there is an integer k∈ [0,n] such that w(m)∈ if and
only if m≤ k.  That is, once the path leaves , it never returns to .
Many of our calculations involve summing over values of this quantity k.

Recall that we assume that there are two possible distributions for V(x), 
called V and Ṽ:

  If x∈, then V(x) has distribution Ṽ.   

  If x∈∖, then V(x) has distribution V.



For a node x with Height(x)≤ n, let 

    Z_n^[x](β)    := ∑_W:x→ (n,·)   e^β V⟨ W⟩

where the sum is over all directed paths W in  from x to some node in the n^th generation,
and the Hamiltonian is 

    V⟨ W⟩ := ∑_y∈ W∖{x} V(y)   .


We shall write
Z_n^[x](β) for whichever model is under
consideration, suppressing other details from the notation.





 §.§ The deterministic model:  Proof of Theorem <ref>.



Recall that we have 𝐏(V=0) = 1 and 𝐏(Ṽ=u) = 1 for this model.  
Then the partition function can be written explicitly as 

    Z^Det_n(β,u) = ∑_k=0^n-1e^kβ u d_1^k(d-d_1)  d^n-k-1+d^n_1e^nβ u .

The free energy, Equation (<ref>), exists because 

    max{ e^nβ ud_1^n,(d-d_1)d^n-1} ≤  Z^Det_n(β,u)
      ≤   (n+1) (max{e^β ud_1,d})^n

which shows that

    f^Det(β,u) = max{β u+log d_1,log d}.

From Equation (<ref>), it follows that the critical curve defined in Equation (<ref>)  is given by

    u^Det_c(β)  = log (d/d_1)/β .






 §.§ The defect branch:  Proof of Theorem <ref>.


First, we shall introduce some notation. For each nonnegative integer m, let S_m be the sum of the
(unshifted) disordered variables along the 
left-most branch of the tree up to the m^th generation, that is,

    S_m  := ∑_k=1^m V_k,1 and  S_0=0.

Recalling the definition in Equation (<ref>), let 

    G^Br_k,n(β)  := ∑ _y∈ D((k,1))e^β V(y)Z_n^[y](β)  for    
      0≤ k<n,
         and
    
      G^Br_n,n(β)    :=  1.

That is, G^Br_k,n(β) is the sum of all contributions by walks from node x=(k,1) up
to height n that do not pass through the node (k+1,1).

Then the partition function can be written as

    Z^Br_n(β,u)   = ∑_k=0^n e^β(uk+S_k)G^Br_k,n(β).



Observe that for each y ∈ D((k,1)), the quantities Z_n^[y](β) and Z^HD_n-k-1(β) 
have the same distribution.  
Moreover, we see that the sum G^Br_k,n(β) is stochastically smaller than 
Z^HD_n-k(β), which by definition means that 

    𝐏(G^Br_k,n(β)  ≥ A)       ≤  𝐏(Z^HD_n-k(β)  ≥ A)
         for every real constant A.


Fix β and u, and 
let ϵ>0 be given. By Equation (<ref>), there exists a constant 
n_o=n_o(ϵ,β)≥ 1 such that 

    𝐄log Z_j^HD(β)  ≤  j(ϕ(β)+ϵ)
       for every integer j≥ n_o.

From Equations (<ref>), (<ref>), and (<ref>), there exist 
n_1=n_1(ϵ,β) and c=c(ϵ) such that for all 
nonnegative integers k and n with
n-k≥ n_1, we have

    𝐏(G_k,n^Br(β)≥ e^(n-k)[ϕ(β)+2ϵ])    ≤ 𝐏(Z_n-k^HD(β)≥ e^(n-k)[ϕ(β)+2ϵ])    
       ≤ 𝐏(log Z_n-k^HD(β)≥𝐄log Z_n-k^HD(β)
         +(n-k)ϵ)    
       ≤  e^-c(n-k)^1/3 .


Let us define the quantities W := max{β(u+μ),ϕ(β)} and

    p_n  := 𝐏( Z^Br_n(β,u)
         ≥   (n+1) e^n(W+3ϵ))   for n≥ 1.


By Equation (<ref>), for every n we obtain 

    p_n     ≤   ∑_k=0^n𝐏(e^β(ku+S_k)G_k,n^Br(β) ≥ 
        e^k(W+ϵ )+nϵe^(n-k)(W+ϵ)+nϵ)
       ≤   ∑_k=0^n𝐏(e^β(ku+S_k)G_k,n^Br(β) ≥ 
        e^k(β(u+μ)+ϵ )+nϵe^(n-k)(ϕ(β)+ϵ)+nϵ)
       ≤    A_n+B_n  ,

where

    A_n     := ∑_k=0^n𝐏(e^β(ku+S_k) ≥ 
        e^k(β(u+μ)+ϵ )+nϵ)   
        and
    
        B_n    := ∑_k=0^n𝐏(G_k,n^Br(β) ≥ 
        e^(n-k)(ϕ(β)+ϵ)+nϵ) .


We shall handle A_n by a standard “large deviation” bound.
Recall that λ(β)=log𝐄[e^β V] and μ=𝐄(V).
For  every t>0 and  every α>0, we have

    𝐏(S_m≥ m(μ+t))  ≤  e^-m[α(μ+t)-λ(α)]for every m≥ 1

(see for example Equation (2.6.2) of <cit.>).
Since λ'(0)=μ, there exists α^*>0 such that α^*(μ+ϵ)-λ(α^*)>0
(this is Lemma 2.6.2 of <cit.>).   Therefore, for every k∈ [1,n], we have

    𝐏(S_k  ≥  k (μ+ϵ+nϵ/kβ))
         ≤  e^-k[α^*(μ+ϵ+nϵ/kβ)-λ(α^*)] <  e^-n(α^*ϵ/β).

Thus, observing that the k=0 summand of A_n equals 0, we have

    A_n    = ∑_k=1^n𝐏(S_k≥ k(μ+ϵ+nϵ/kβ) )
              ≤   n  e^-n(α^*ϵ/β)for all n≥ 1.

For B_n, let m_n=⌊ n-√(n)⌋.  
Using Equations (<ref>) and (<ref>) and Markov's inequality, 
we deduce that if √(n)>n_1(ϵ,β), then  

    B_n    ≤   ∑_k=0^m_n𝐏(G_k,n^Br(β)≥ e^(n-k)(ϕ(β)+ϵ)+
        (n-k)ϵ)
             +  ∑_k=m_n+1^n𝐏(Z_n-k^HD(β)≥ e^(n-k)(ϕ(β)+ϵ)
       +nϵ)
       ≤   ∑_k=1^m_n e^-c(n-k)^1/3 + ∑_k=m_n+1^n𝐄 Z_n-k^HD(β) /e^(n-k)[ϕ(β)+ϵ]+nϵ
       ≤    n e^-c n^1/6+∑_k=m_n+1^ne^(n-k)(λ(β)+log d)/
         e^(n-k)ϕ(β) e^nϵ
       ≤    n e^-c n^1/6 +(√(n)+1)  e^-nϵ e^√(n) (λ(β)+log d-ϕ(β))

(where the last inequality holds because ϕ(β)≤λ(β)+log d). 



Recalling that p_n ≤  A_n+B_n, we see from Equations (<ref>) and (<ref>)
that ∑_n=1^∞p_n converges.  Therefore, the Borel-Cantelli Lemma tells us that

    𝐏(  1/nlog Z_n^Br(β,u)  ≥ 1/nlog(n+1) +
       W+3ϵ  for infinitely many values of n)  =  0  .

 Therefore lim sup_n→∞1/nlog Z_n^Br(β,u)
 ≤ W+3ϵ almost surely.  Since ϵ can be made arbitrarily close to 0, we obtain
 
    lim sup_n→∞1/nlog Z_n^Br(β,u)  ≤  W     a.s.

 
Note also that 

    e^β u n +S_n+e^β V((1,2)) Z_n^[(1,2)](β)  ≤  Z^Br_n(β,u)  .

By the Strong Law of Large Numbers (applied to S_n), and Equation (<ref>) (recalling
that Z_n^[(1,2)](β) and Z_n-1^HD(β) have the same distribution),
we conclude that 

    W=max{β(u+μ),ϕ(β)} ≤ lim inf_n→∞1/nlog Z^Br_n(β,u)
       a.s.

This completes the proof of Theorem <ref>.



If the disorder distribution is normal with mean μ and variance σ^2, then β_c=√(2 log d)/σ and

    u^Br_c(β)={[ 1/2σ^2β+log d/β    if   β < β_c;      σ√(2log d)     if   β≥β_c. ].


 Let V be a general disorder distribution with mean zero and variance σ^2. Then λ(β)∼1/2σ^2β^2 as β→ 0. Therefore, u^Br_c(β)∼1/2σ^2β+log d/β as β→ 0.
 


More generally, our proofs can be easily modified to show that for the case d_1=1 
and V(x) d= Ṽ for all x∈, we have 

    lim_n→∞1/nlog Z_n^[0](β)    = max{β𝐄(Ṽ),ϕ(β)}a.s.

This reduces to Theorem <ref> in Case I, where 𝐄(Ṽ) = u+𝐄(V).





 §.§ The defect subtree:  Proof of Theorem <ref> and some auxiliary results.




For the model with a non-disordered defect subtree, it is not obvious how to prove that the 
limiting free energy exists almost surely.

Therefore, we make the following definitions:

    f^ST(β,u)      :=    lim sup_n→∞1/n log Z^ST_n(β,u) 
    f^ST(β,u)      :=    lim inf_n→∞1/n log Z^ST_n(β,u) .
   
By the Kolmogorov zero-one law, 
f^ST(β,u) and f^ST(β,u) are constant almost surely,
so we shall treat f^ST and f^ST as deterministic functions.
If f^ST(β,u) = f^ST(β,u), then we define f^ST(β,u)
to be the common value; in other words, 

    f^ST(β,u)  := lim_n→∞1/n log Z^ST_n(β,u)
         if this limit exists.

 
We can now formalize the definition of the three phases that we introduced in Section 
<ref>.


We define the three phases as follows:

    R_FP    :=    {(β,u): f^ST(β,u)=β u+log d_1}(fully pinned)
     R_D    :=    {(β,u): f^ST(β,u)=ϕ(β)}(depinned)
     R_PP    :=    {(β,u): f^ST(β,u)>max{ϕ(β),β u+log d_1} }(partially pinned).

 
 
 Implicit in the definitions is that the limiting free energy of Equation (<ref>) must exist 
 for every point of R_FP and R_D.
 
Let's recall the definitions of the functions F and J defined in section <ref>:  
 
    F(β)       =    1/β (λ(β)+log d -log d_1)   
    
          
          J(β)     =    1/β( ϕ(β)-log d_1-
          [λ(β)+log d-ϕ(β)]log d_1/λ(2β)-2λ(β)-log d) .



 
    
Our characterization of the phases in Theorem <ref> is not complete for β>β_c when
u is between F(β_c) and J(β).
However, the following proposition, combined with Theorem <ref>,
shows that the partially pinned phase is nonempty, and indeed it
contains points (β,u) with β arbitrarily close to β_c.

 

For every β>β_c,

    F(β_c)  = J(β_c) <  J(β)  < ϕ(β)-log d_1/β < F(β).




The proof of  Proposition <ref> appears at the end of this section,
immediately before the proof of Theorem <ref>. We shall first prove some preliminary results.
 

   
For every β>0 and every u,

    lim inf_n→∞1/n log Z^ST_n(β,u)
        ≥ max{ϕ(β), β u+log d_1}   a.s.



Before we prove Proposition <ref>, observe that in any path W from the root  to generation n, there is a node x in W such that 
the part of W from  to x is contained in , and the rest of W is 
outside .
Writing k to represent the generation of x, we see that

    Z^ST_n(β,u)   = ∑_k=0^n-1∑_x∈T̃:  Height(x)=k∑_y∈ D(x) e^β k u e^β V(y) Z_n^[y](β)    +   d_1^ne^β nu.

(The rightmost term corresponds to those paths that never leave , i.e. k=n.)

We shall use Equation (<ref>).  Specifically, for y∈ D(), we have

    Z^ST_n(β,u)   ≥   e^β V(y) Z_n^[y](β) 
         and    
      Z_n^[y](β)     Z_n-1^HD(β)  .

Then by Equation (<ref>), we have

    lim inf_n→∞1/n log Z^ST_n(β,u)
        ≥ ϕ(β)     a.s.

Also, since Z^ST_n(β,u)   ≥ d_1^ne^β nu, we have 

    lim inf_n→∞1/n log Z^ST_n(β,u)  
        ≥ log d_1   + β u a.s.

The proposition follows from inequalities (<ref>) and (<ref>).

We introduce the following notation:  for 0≤ k< n, 

    G^ST_k,n(β)   := ∑_x∈T̃:  Height(x)=k ∑_y∈ D(x) e^β V(y) Z_n^[y](β)   .


(Observe that in the case d_1=1, the above would reduce to G_k,n^Br(β) as
defined in Equation (<ref>).)
Then we see from Equation (<ref>) that

    Z^ST_n(β,u)    = ∑_k=0^n-1 e^β ku G^ST_k,n(β)    +   d_1^ne^β nu,

which is the analogue of Equation (<ref>).
Since G^ST_k,n(β) is a sum of d_1^k(d-d_1) independent copies of 
e^β V^*Z_n-k-1^HD(β) (where V^* is a copy of V, independent of everything else), 
we have

    𝐄(G^ST_k,n(β))    =   d_1^k(d-d_1)d^n-k-1 e^(n-k)λ(β)and

 
    Var(G^ST_k,n(β))    =   d_1^k(d-d_1)
           Var(e^β V^* Z^HD_n-k-1(β))  .
      
      
      
      

    
For every β>0 and every u, 

    lim sup_n→∞1/n log Z^ST_n(β,u)  
        ≤ max{λ(β)+log d, β u+log d_1}a.s.



For given (β,u), let M  = max{λ(β)+log d, β u+log d_1}.   Fix C>M.
By Equation (<ref>), we have

    𝐏(  1/nlog Z^ST_n(β,u)  > C)       = 𝐏( Z^ST_n(β,u) > e^Cn)    
       ≤ ∑_k=0^n-1𝐏(e^β ku G^ST_k,n(β) > e^Cn/n+1)
              +  1(d_1^n e^β nu > e^Cn/n+1)  .

The rightmost term in Equation (<ref>) is zero for all sufficiently large n, since
C> β u +log d_1.  For 0≤ k≤ n-1, we have

    𝐏   (e^β ku G^ST_k,n(β) > e^Cn/n+1)   
        
       ≤ e^β ku 𝐄(G^ST_k,n(β))/e^Cn/(n+1)(by Markov's Inequality)
        <  e^β ku d_1^k d^n-k e^(n-k)λ(β)/e^Cn/(n+1)(by Equation  (<ref>))
        =   (n+1) 
                (e^λ(β)+log d)^n-k(e^β u+log d_1)^k/e^Cn
       ≤  (n+1)
            e^n(M-C) .
     
Hence, for large n, 

    𝐏(  1/nlog Z^ST_n(β,u)   > C)    ≤    
         n(n+1)  
        e^n(M-C) .

Since M-C<0, the Borel-Cantelli Lemma shows that, with probability 1, there are only finitely
many values of n for which 1/nlog Z^ST_n(β,u)   > C.  This proves that

    lim sup_n→∞1/n log Z^ST_n(β,u)  ≤ C  a.s.

Since C can be 
arbitrarily close to M, Proposition <ref> follows.




Since ϕ(β)=λ(β)+log d for β≤β_c, 
Propositions <ref> and <ref> immediately imply the following.


   
For every β≤β_c, 

    lim_n→∞1/n log Z^ST_n(β,u)  
         =  max{ϕ(β), β u+log d_1} .

 In particular, the limit exists almost surely.


Since ϕ(β) < λ(β)+log d for β > β_c (Equation (<ref>)), 
we must ask whether
the conclusion of Corollary <ref> holds for all values of β.  We shall show 
that the answer is no.
This is interesting because it tells us that the dominant terms in the partition function 
are neither the walks that spend almost all their time in the defect subtree nor the walks
that spend hardly any time in the defect subtree.  This is in direct contrast to the case of 
a defect branch (d_1=1), which we examined in Section <ref>.




The next lemma will be needed for an application of Chebychev's Inequality.

   
(a) For every β≥ 0, the limit 

    Θ ≡Θ(β)  := lim_n→∞(  
        Var(e^β V^* Z^HD_n(β)))^1/n

exists and equals max{d^2e^2λ(β),d e^λ(2β)}.

(b)  For every β>β_c, we have Θ =  d e^λ(2β).


(a) Let us write Y=e^β V^* and Z_n =Z_n^HD(β). Recall that V^* is a copy of V, independent of everything else. Since Y and 
Z_n are independent, it is easy to see that 
Var(YZ_n) = 𝐄(Z_n^2)Var(Y)+(𝐄(Y))^2Var(Z_n), and hence that

    𝐄(Z_n^2)Var(Y)   ≤ Var(YZ_n)   ≤ 𝐄((YZ_n)^2)   = 𝐄(Y^2)𝐄(Z_n^2)  .

Since Y does not depend on n, we deduce that  
Θ =lim_n→∞𝐄(Z_n^2)^1/n if this limit exists.  
Using this observation,  
part (a) follows from the following calculation:

    𝐄[(Z^HD_n(β))^2]   =   ∑_W:→ (n,·)∑_W':→ (n,·)𝐄[e^β V⟨ W⟩+β V⟨ W'⟩]
       =   d^n(∑_k=0^n-1(d-1)d^n-k-1(e^λ(2β))^k (e^2λ(β))^n-k
           +e^nλ(2β))
        =     d (d-1) e^2λ(β)∑_k=0^n-1(de^λ(2β))^k
       (d^2e^2λ(β))^n-1-k +(de^λ(2β))^n   
        =        d(d-1)  
          e^2λ(β) (de^λ(2β))^n-(d^2e^2λ(β))^n/
        de^λ(2β)-d^2e^2λ(β)
           +(de^λ(2β))^n.


(b) Fix β>β_c, and assume that d^2e^2λ(β)>de^λ(2β).  
That is, log d> λ(2β)-2λ(β). 
By the Mean Value Theorem, we know that λ(2β)=λ(β)+βλ'(β̃) 
for some β<β̃<2β.
Recalling Equation (<ref>) and Lemma <ref>, we find that

    f(β)   >   λ(β)+λ(2β)-2λ(β)-βλ'(β)
       =   β (λ'(β̃)-λ'(β))
       >   0    (since λ”>0).
 
This contradicts the fact that f(β)<0 for β∈ (β_c,∞).
Therefore d^2e^2λ(β)≤ de^λ(2β).  Part (b) follows.


The following lemma plays an important role in proving that the partially pinned phase is not empty for β>β_c. 


Fix β>0.  Let t be a real number in (0,1) such that 

    (Θ/d^2e^2λ(β))^1-t <   d_1^t  .

Then for every u,

    lim inf_n→∞1/n log Z^ST_n(β,u)  
        ≥   (λ(β)+log d)(1-t)  +    (β u+log d_1)t  a.s.




   
   Observe that Equation (<ref>) holds when t is close enough to 1.


Before we prove Lemma <ref>, we shall show how it can be used.


   
Assume β>β_c.  Then the strict inequality 

    lim inf_n→∞1/n log Z^ST_n(β,u) 
         > max{ϕ(β), β u+log d_1}a.s.
   
holds  if either of the following  hold:


(a)  ϕ(β)  = β u +log d_1; or


(b)  λ(β)+log d  > β u +log d_1  >  ϕ(β).


In particular,  (β,u) is in the partially pinned phase R_PP if
(ϕ(β)-log d_1)/β ≤ u < F(β).


The proof of Theorem <ref> will also use Lemma <ref> to prove that 
the inequality (<ref>) also holds if β u+log d_1 is smaller than, but 
sufficiently close to, ϕ(β).


 
Let M = max{ϕ(β), β u+log d_1}.

(a)  In this case, M = ϕ(β)  = β u +log d_1.  Since ϕ(β)<λ(β)+log d
(by Equation (<ref>)), 
we see that the right side of Equation (<ref>) is strictly greater than M for every t 
in the interval (0,1).  By Remark <ref>, the result follows.

(b)  In this case, λ(β)+log d  >  M  = β u+log d_1.  As in part (a), the result 
follows.




For 0≤ k< n, define the event 

    A[k,n]   :=    {|G^ST_k,n(β)-𝐄(G^ST_k,n(β)) |  ≥ 1/2 𝐄(G^ST_k,n(β))   } .

By Chebychev's Inequality, and Equations (<ref>) and (<ref>), we have 

    𝐏( A[k,n])    ≤   4 Var(G^ST_k,n(β))/(𝐄(G^ST_k,n(β))^2
        =     4 Var(e^β V^* Z_n-k-1^HD(β))/d_1^k(d-d_1) 
                d^2(n-k-1) e^2(n-k)λ(β)

 
 Next, we let k=k(n) be an integer-valued function of n with the property that
 
    lim_n→∞k(n)/n  =   t
    
 where t is given in the statement of the Lemma.  Then
 
    lim sup_n→∞𝐏( A[k(n),n])^1/n ≤ Θ^1-t/d_1^t (de^λ(β))^2(1-t) <  1

 where the final inequality is a consequence of Equation (<ref>).   Therefore
𝐏( A[k(n),n]) decays to 0 exponentially rapidly in n, and 
 hence the Borel-Cantelli Lemma shows that (with probability 1) A[k(n),n]  occurs for
 only finitely many values of n. 
      
  Observe that G^ST_k,n(β) > 𝐄(G^ST_k,n(β))/2   on A[k,n]^c       
  (where ^c denotes complement).  Therefore

    G^ST_k(n),n(β)     ≥ 1/2 𝐄(G^ST_k,n(β)) 
         1( A[k(n),n]^c)  .

The final conclusion of the previous paragraph, together with Equation (<ref>), shows
that 

    lim inf_n→∞ G^ST_k(n),n(β)^1/n ≥   
              d_1^t(de^λ(β))^1-ta.s.
   
Finally, Equation (<ref>)  implies that Z^ST_n(β,u)  ≥ 
e^β k(n)uG^ST_k(n),n(β).
Lemma <ref> follows immediately from this and  Equation (<ref>).



We define the critical curve as

    u^ST_c(β):=inf{u∈ℝ: f^ST(β,u)>ϕ(β)}.


Then we have the following.

  
  Assume β≥β_c and u≤Ψ, where

    Ψ := 1/β_c(λ(β_c)+log d-log d_1)  .

Then

    lim_n→∞1/nlog Z^ST_n(β,u)     = ϕ(β)   a.s.

That is, u^ST_c(β) ≥ Ψ for all β≥β_c.



Fix β≥β_c and u≤Ψ.   Let ϵ>0 and let

    C := ϕ(β) + ϵ = β/β_c(λ(β_c)+log d)   + ϵ .

Our approach is similar to the proof of Proposition <ref>.  As in Equation (<ref>),
we have  

    𝐏(  1/nlog Z_n^ST(β,u)   > C)  = 𝐏(  ∑_k=0^n e^β kuG^ST_k,n(β)  > e^Cn)
              
       ≤ ∑_k=0^n-1𝐏(e^β ku G^ST_k,n(β) > e^Cn/n+1)
              +  1(d_1^n e^β nu > e^Cn/n+1)  .

Since β u+log d_1 ≤ βΨ+(β/β_c)log d_1 < C, the final term in Equation (<ref>) is 0 for all
sufficiently large n.

Since β≥β_c, we have G^ST_k,n(β)^1/β ≤  
G^ST_k,n(β_c)^1/β_c by Lemma 5 of <cit.>,  and thus

    𝐏   (e^β ku G^ST_k,n(β) > e^Cn/n+1)
        
       ≤ 𝐏(G^ST_k,n(β)^β/β_c > e^Cn-β ku/n+1)
       
        =  𝐏(G^ST_k,n(β) > e^(β_c/β)Cn-β_c ku/(n+1)^β_c/β)
            
       ≤ (n+1)^β_c/βe^β_c ku 𝐄 (G^ST_k,n(β))/e^(β_c/β)Cn(by Markov's Inequality)
        =  (n+1)^β_c/βe^β_c ku d_1^k (d-d_1)d^n-k-1
         e^(n-k)λ(β_c)/e^(β_c/β)Cn(by Equation  (<ref>))
            
       ≤  (n+1)    (e^λ(β_c)+log d/e^Cβ_c/β)^n-k(e^β_c u+log d_1/e^Cβ_c/β)^k   .

Since

    Cβ_c/β = λ(β_c)+log d +β_cϵ/β =  β_cΨ  +log d_1 + β_cϵ/β ≥ β_cu+log d_1 +β_cϵ/β ,

we conclude that for large n,

    𝐏(  1/nlog Z_n^ST(β)   > C)    ≤ 
     n(n+1) e^-nϵβ_c/β .

It follows from the Borel-Cantelli Lemma that
lim sup_n→∞1/nlog Z_n^ST(β,u)  ≤ ϕ(β)+ϵ with
probability 1.  Since this holds for every positive ϵ, we can combine this with
Proposition <ref> to complete the proof of Equation (<ref>).


We are now ready to prove the main results of this section.


  
Note that λ(β_c)+log d=ϕ(β_c) and hence F(β_c)=J(β_c). 
Fix β> β_c.  We have λ(β)+log d>ϕ(β) (by Equation (<ref>)) 
and 

    λ(2β)-2λ(β)-log d>0

by Lemma <ref>(a,b).
Hence the second and third inequalities follow.

It remains to prove the first inequality in the proposition.  
Since λ is strictly convex, we can consider slopes of secant lines to obtain

    λ(2β)-λ(β)/β > λ(β)-λ(β_c)/β-β_c .
   
Now replace ϕ(β) by β/β_c(λ(β_c)+log d) in the definition
of J(β) in Equation (<ref>), obtaining

    J(β)     =    λ(β_c)+log d/β_c - log d_1/β( 1 + λ(β)+log d-ϕ(β)/λ(2β)-2λ(β)-log d)   
        =     F(β_c)+log d_1/β_c -
           log d_1/β( λ(2β)-λ(β)-ϕ(β)
                 /λ(2β)-2λ(β)-log d)  .

Now, some algebra gives

    J(β)-F(β_c)/log d_1    =    1/β_c  -1/β( λ(2β)-λ(β)-(β/β_c)[λ(β_c)+log d]
                 /λ(2β)-2λ(β)-log d)     
        =    (β-β_c)[λ(2β)-λ(β)] -β[λ(β)-λ(β_c)]
          /ββ_c  [λ(2β)-2λ(β)-log d]
        >     0   (by Equations (<ref>) and (<ref>)).

Therefore J(β)>F(β_c), and the proof is complete.



We start with the observation that every point (β,u) (with β>0) is in at least one
of R_FP or R_D or R_PP.  
To see this, suppose  (β,u)∉ R_PP. Then 
max{ϕ(β),β u+log d_1}≥f^ST(β,u)≥f^ST(β,u).
But f^ST(β,u)≥max{ϕ(β),β u+log d_1} by Proposition <ref>,
so the limit f^ST(β,u) exists  and equals  β u+log d_1 or ϕ(β).  That is,
(β,u)∈ R_FP∪ R_D.


To prove part (a),  fix  β≤β_c.
By    Corollary <ref>, the limiting free energy exists and is given by

    f^ST(β,u)     = max{ϕ(β),β u+log d_1} .

First assume u≥ F(β). This is equivalent to  
β u+log d_1 ≥λ(β)+log d.  Since 
ϕ(β)=λ(β)+log d, we obtain f^ST(β,u)=β u+log d_1, and 
hence (β,u)∈ R_FP.  Similarly, the assumption u≤ F(β) leads
to β u+log d_1≤λ(β)+log d=ϕ(β), and hence f^ST(β,u)=ϕ(β),
i.e. (β,u)∈ R_D.

Now we shall prove part (b).  Fix β >β_c.

First assume u≥ F(β).  Then β u+log d_1≥λ(β)+log d ≥ϕ(β)
(the second inequality is from Equation (<ref>)).  By Propositions <ref>
and <ref>, we have

    f^ST(β,u)  ≥ β u+log d_1  ≥ f^ST(β,u)   ,

which shows that f^ST exists and equals β u+log d_1.  Therefore (β,u)∈ R_FP.


Next, assume that u≤ F(β_c).  Then Proposition <ref> says that
(β,u)∈ R_D, since F(β_c)=Ψ.

Next, assume that J(β)<u<F(β).   Let 

    t^*  ≡  t^*(β)      =    λ(2β)-2λ(β)-log d/log d_1+λ(2β)-2λ(β)-log dand
    
             L(β,t)      =    ϕ(β)-(1-t/t)(λ(β)+log d-ϕ(β))
                for t≠ 0.

By Lemma <ref>(a,b), we know that λ(2β)-2λ(β)-log d>0,
which implies that t^*∈ (0,1). Also, observe  that 

    J(β)    = L(β,t^*)-log d_1/β .

Recalling from Lemma <ref>(b) that Θ =de^λ(2β), it is easy to 
show that 

    (Θ/d^2e^2λ(β))^1-t^* =   d_1^ t^* ,

and that the inequality of Equation (<ref>) holds whenever t^*<t<1.

Still assuming  J(β)<u<F(β), we consider two possible cases:

(i)   ϕ(β)≤β u+log d_1, or


(ii)   β u+log d_1<ϕ(β).

If case (i) holds, then Proposition <ref> says that
(β,u)∈ R_PP.  So we shall assume that case (ii) holds.  
Since 
(L(β,t^*)-log d_1)/β<u, we can choose t∈ (t^*,1) such that 
(L(β,t)-log d_1)/β<u.   From simple algebra, it follows that, for this value of t, 
the right hand side
of Equation (<ref>) is strictly greater than ϕ(β), which 
in turn equals max{ϕ(β),β u+log d_1} in case (ii).  
For this choice of t, Lemma <ref> shows
that  (β,u)∈ R_PP.


Finally, assume that F(β_c)<u≤ J(β).  
Since ϕ(β)<λ(β)+log d  (by Equation (<ref>)), 
we see from Equation (<ref>) that L(β,t^*)<ϕ(β).
We then have u≤ J(β)<(ϕ(β)-log d_1)/β (by Equation (<ref>)), and hence
β u+log d_1<ϕ(β).  Therefore, by Proposition <ref>, 
we have f^ST(β,u)≥ϕ(β)>β u+log d_1.  Hence (β,u) is not 
in R_FP, so it must be in R_D or R_PP by the first observation of the
present proof.






§ DISCUSSION OF THE RESULTS AND SOME FUTURE RESEARCH DIRECTIONS


In this section, we will discuss our results and also mention some possible future research directions.



 §.§ Polymers on disordered trees with a shifted-disordered defect subtree

Let's assume that Ṽ(x)=V(x)+u for x∈, so that  the Hamiltonian is given by  

    V⟨ W⟩ := ∑_y∈∩ (W∖{}) (V(y)+u)+∑_y∈ (∖)∩ (W∖{}) V(y) .

Note that the localized microscopic defect was non-random in the model introduced in section <ref>. 

We denote the partition function of this model by

    Z̃_n^ST(β,u):=Z^[]_n(β).

Then for u≥ 0, we have

    lim inf_n→∞1/nlogZ̃_n^ST(β,u) ≥max(ϕ(β),β u+ϕ̃(β))  a.s.

where 

    ϕ̃(β)={[         λ(β) +log d_1           if   β <β_c; β/β_c(λ(β_c)+log d_1)           if   β≥β_c. ].

To see that the left side of the Equation (<ref>) is greater than ϕ(β),  we restrict the partition function to ∖; and to see that it is greater than β u+ϕ̃(β),
we restrict the partition function to .

We don't yet know much more about this general model but it
is reasonable to suspect a  phase diagram similar to Figure <ref>.



 §.§ Directed polymers on disordered integer lattice ℤ^d with a defect line
 

The 1+d dimensional lattice version of the directed polymer in a random environment is formulated as follows.




The polymer configurations are represented by the directed paths of the simple symmetric random walk (SSRW) {(j, S_j)}_j=1^n in ℕ×ℤ^d.  The disordered random environment is given by i.i.d. random variables {v(i,x) : i ≥ 1, x ∈ℤ^d} with law denoted by 𝐏 satisfying 

    λ(β)=log𝐄[e^β v(i,x)]< ∞  for all  β∈ℝ.

The Hamiltonian of the model is given by

    H_n(S):=∑_j=1^nv(j, S_j)

and Z_n(β):=∑_Se^β H_n(S)
denotes the partition function where the sum is overall SSRW paths of length n with S_0=0. The free energy of the model is defined as

    f(β):=lim_n→∞1/nlog Z_n(β).

The existence of the free energy is first proven by Carmona and Hu <cit.> for the Gaussian environment and then for any distribution which satisfies the exponential moment condition in Equation (<ref>) by Comets et al. in <cit.>. There is no explicit expression for the free energy for the lattice case as opposed to the tree case as in Equation (<ref>).

The first rigorous mathematical work on the directed polymers in 1+d dimensions was done by Imbrie and Spencer <cit.>, proving that in dimension d≥ 3 with Bernoulli disorder and small enough β, the end point of the polymer scales as n^1/2, i.e. the polymer is diffusive. Later, Bolthausen <cit.> extended this to a central
 limit theorem for the end point of the walk, showing that the polymer behaves almost as if the disorder were absent. In the same paper, Bolthausen also introduced the nonnegative martingale M_n(β)=Z_n(β)/𝐄[Z_n(β)] and observed that for the positivity of the limit M(β)=lim_n→∞M_n(β), there are only two possibilities, 𝐏(M(β)>0)=1, known as weak disorder, or 𝐏(M(β)=0)=1, known as strong disorder.
Comets and Yoshida <cit.>, showed that there exists a critical value β_c∈ [0,∞], with 
β_c =0 for d=1,2 and  0<β_c ≤∞ for d ≥ 3, such that 
𝐏(M(β)>0)=1  if β∈{0}∪(0,β_c) and 𝐏(M(β)=0)=1  if β >β_c. In particular, for the 1+1 dimensional case, disorder is always strong. It is not known whether β_c belongs to the weak disorder or strong disorder phase for the lattice version, whereas we know that β_c belongs to the strong disorder phase for the tree case.

In the 1+1 dimensional case, the localized microscopic defect is incorporated to the model by modifying the Hamiltonian as follows:


    H^u_n(S):=∑_j=1^n(v(j, S_j)+u1_S_j=0)

and Z_n(β,u):=∑_Se^β H^u_n(S)
denotes the partition function. The free energy and the critical curve of the model are defined as

    f(β,u):=lim_n→∞1/nlog Z_n(β,u)


    u_c(β):=inf{u≥0: f(β,u)>f(β,0)}.

For the existence of the limit in Equation (<ref>) and its self-averaging property, see <cit.>.
As we discussed in section <ref>, the question of whether u_c(β)>0 or not for some range of β is still an open question. One of the reasons why it is not easy to solve this question rigorously is that a nice decomposition, such as in Equation (<ref>), is not available for the partition function of the lattice model.





 §.§ Directed polymers on disordered hierarchical lattices with defect substructure
 
The directed polymers on disordered hierarchical lattices were first introduced and studied in the 
physics literature by Derrida and Griffiths <cit.>, and Cook and Derrida <cit.> for the bond disordered case, and then rigorously by Lacoin and Moreno <cit.> for the site disordered case. The hierarchical lattices are usually generated by an iterative rule as described for the diamond lattice: The first generation, D_0, consists of two sites, labeled as A and B, with one bond. In the next generation, D_1, the bond is replaced by a set of four bonds, and then in each step, each bond is replaced by such a set of four bonds to form the next generation, see Figure <ref>.
For more general hierarchical lattices, the generation D_n+1 is obtained by replacing each bond in the generation D_n by b branches of s bonds. The directed paths in D_n linking the sites A and B represent the polymer configurations. The disorder is introduced in the model by assigning independent random variables from a distribution to each site. The Hamiltonian of the model, partition function, and free energy are defined as in lattice and tree version of the model, and the martingale defined by the normalized partition function separates two phases as weak and strong disorder depending on the lattice parameters b, s and the inverse temperature β, for the details see <cit.>. In <cit.>, in particular, they prove that the free energy exists almost surely and it is a strictly convex function of β which holds also for the directed polymer on ℤ^d but not on the tree for β>β_c. As noted in <cit.>, this  fact is related to the “correlation structure” of the models as two directed paths on ℤ^d and hierarchical lattice can re-intersect after being separated at some point which is not the case for the tree model. Among these three directed polymer models, only the tree version is exactly solvable, that is, there is an explicit expression for the free energy. 

The localized microscopic defect is incorporated to the model by enhancing the disorder variables along a single directed path from A to B with a fixed potential u, see Figure <ref>. The main question is determining whether the critical point for the extra potential is zero or not depending on the model parameters, inverse temperature β, and lattice parameters b,s; that is whether u_c(β,b,s)=0 or not for some β, b, s. This problem was studied in <cit.> by using Migdal-Kadanoff renormalization group method but the results lack the rigor of formal mathematical proofs.


N. Madras was supported in part by a Discovery Grant from NSERC of Canada.



§ REFERENCES

99

AlYil Alexander, K. S. and  Yıldırım, G. Directed polymers in a random environment with a defect line. Electronic Journal of Probability, 20, no 6, 20 pp, 2015.

BK
Balents, L. and Kardar, M. Delocalization of flux lines from extended
  defects by bulk randomness. Europhys. Lett., 23, 503–509, 1993.


BK1
Balents, L. and Kardar, M. Disorder-induced unbinding of a fiux line from an extended defect. Physical Review B, 49, 18, 13030(19), 1994.

BSS
Basu, R., Sidoravicius, V. and Sly, A. Last passage percolation with a defect
  line and the solution of the slow bond problem. https://arxiv.org/abs/1408.3464. 2014
  
BBDD Beaton, N., Bousquet-Mélou, M., De Gier, J., Duminil-Copin, H. and Guttmann, A. The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+√(2). Communications in Mathematical Physics, 326(3), 727–754, 2014.

BSSV
Beffara, V., Sidoravicius, V., Spohn, H. and Vares, M. E. Polymer pinning in a random medium as influence percolation.Dynamics and Stochastics, IMS Lecture Notes Monogr. Ser. 48, Inst. Math. Statist., Beachwood, OH, 2006. 

BSV
Beffara, V., Sidoravicius, V. and Vares, M. E. Randomized polynuclear growth with a columnar defect. Probab. Theory Rel. Fields, 147,
  no. 3-4, 565–581, 2010.
  
Biggins Biggins, J. D. Martingale convergence in the branching random walk. J. Appl. Probability, 14, no. 1, 25–37, 1977.

Bolt Bolthausen, E. A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys., 123, no. 4, 529–534, 1989.

BSL
Budhani, R. C., Swenaga, M. and Liou, S. H.Giant suppression of flux-flow
  resistivity in heavy-ion irradiated TL_2BA_2Ca_2Cu_3O_10 films:
  Influence of linear defects on vortex transport. Phys. Rev. Lett., 69, 3816–3819, 1992.
  
BPP Buffett, E., Patrick, A. and Pulé, J. V. Directed polymers on trees: a martingale approach. J. Phys. A: Math. Gen., 26, 1823–1834, 1993.  

CaHu1 Carmona, P. and Hu, Y. On the partition function of a directed polymer in a random environment. Probab. Theory Rel. Fields, 124, 431–457, 2002. 

CMWT
Civale, L., Marwick, A. D., Worthington, T. K., Kirk, M. A., Thompson, J. R., Krusin-Elbaum, L., Sum, Y., Clem, Y. R. and Holtzberg, F. Vortex
  confinement by columnar defects in YBa_2Cu_3O_7 crystals: Enhanced pinning at
  high fields and temperatures. Phys. Rev. Lett., 67, 648–652, 1991.


Comets Comets, F. Directed Polymers in Random Environment. Lecture notes for a workshop on Random Interfaces and Directed Polymers, Leipzig, 2005.


Comets1
  Comets, F.
  Directed polymers in random environments. Lecture notes from the 46^th Probability Summer School held in Saint–Flour, Springer–Verlag, 2016.
  
CSY1 Comets, F., Shiga, T. and Yoshida, N. Directed polymers in a random environment: path localization and strong disorder. Bernoulli, 9, no. 4, 705–723, 2003.

CY Comets, F. and Yoshida, N. Directed polymers in a random environment are diffusive at weak disorder. Ann. Probab., 34, (5) 1746–1770, 2006.

CoDer Cook, J. and Derrida, B. Polymers on Disordered Hierarchical Lattices: A Nonlinear Combination
of Random Variables. J. Stat. Phys., 57, 1/2, 89–139, 1989.

denH den Hollander, F. Random polymers. Lectures from the 37^th Probability Summer School held in Saint-Flour. Springer–Verlag, 2009.

DerSp Derrida, B. and Spohn, H. Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys., 51, no. 5–6, 817–840, 1988.


DerGr Derrida, B. and Griffiths, R. B. Directed polymers on disordered hierarchical lattices. Europhys.
Lett., 8, 2, 111–116, 1989.



Durrett Durrett, R. Probability: Theory and Examples. 4^th edition.  Cambridge Series in Statistical and Probabilistic Mathematics, 31. Cambridge University Press, Cambridge, 2010.

Giacomin Giacomin, G. Disorder and critical phenomena through basic probability models. École d'Été de Probabilités de Saint-Flour XL, Lecture Notes in Mathematics, Springer, 2010. 

 Giacomin1 Giacomin, G. Random polymer models. Imperial College Press, World Scientific.  2007.
 


Hamm1 Hammersley, J. M. Critical phenomena in semi-infinite systems.
In Essays in Statistical Science.
J. Appl. Prob., 19A, 327–331, 1982.

 Hamm2 Hammersley, J. M., Torrie, G. M., and Whittington, S. G. 
 Self-avoiding walks interacting with a surface.
J. Phys. A: Math. Gen., 15, 539–571, 1982.

 
HHR
Hansen, A., Hinrichsen,  E. L. and Roux, S. Roughness of crack interfaces.
  Phys. Rev. Lett., 66, 2476–2479, 1991.


HuShi Hu, Y. and Shi, Z. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab., 37, no. 2, 742–789, 2009.

HH
Huse, D. A. and Henley, C. L. Pinning and roughening of domain wall in
  Ising systems due to random impurities. Phys. Rev. Lett., 54,
  2708–2711, 1985.

HN
Hwa, T. and Nattermann, T. Disorder-induced depinning transition. Phys.
  Rev. B, 51, no. 1, 455–469, 1995.
  
IS
Imbrie, J. Z., Spencer, T. Diffusion of directed polymers in a random
  environment. J. Statist. Phys., 52, no. 3-4, 609–626, 1988.
  
KahPey Kahane, J. P.  and Peyriére, J. Sur certaines martingales de Benoit Mandelbrot. Advances in Math., 22, no. 2, 131–145, 1976.  

K2
Kardar, M. Depinning by quenched randomness. Phys. Rev. Lett., 55, 2235–2238, 1985.

KPZ
Kardar, M., Parisi, G. and Zhang, Y. C. Dynamic scaling of growing
  interfaces. Phys. Rev. Lett., 56, no. 9, 889–892, 1986.



LaMo Lacoin, H. and Moreno, G. Directed polymers on hierarchical lattices with site disorder. Stochastic Processes and their Applications., 120, 467–493, 2010. 

 Madras Madras, N. 
 Location of the adsorption transition for lattice polymers.
J. Phys. A: Math. Theor., 50, 064003, 2017.



  
MO Morters, P. and Ortgiese, M. Minimal supporting subtrees for the free energy of polymers on disordered trees. J. Math. Phys. 49, no. 12, 125203, 21 pp, 2008. 


Nel
Nelson, D. R. Vortex entanglement in high-t_c superconductors.
  Phys. Rev. Lett., 60, 1973–1976, 1988.

 Rubin Rubin, R. J.
 Random-walk model of chain polymer adsorption at a surface.
J. Chem. Phys., 43, 2392–2407, 1965.



TL
Tang, L. H. and Lyuksyutov, I. F. Directed polymer localization in a
  disordered medium. Phys. Rev. Lett., 71, 2745–2748, 1993.



