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Continuous Model for Homopolymers

We consider the model for the distribution of a long homopolymer in a potential field. The typical shape of the polymer depends on the temperature parameter. We show that at a critical value of the temperature the transition occurs from a globular to an extended phase. For various values of the temperature, including those at or near the critical value, we consider the limiting behavior of the polymer when its size tends to infinity.


Introduction
The goal of this paper is to analyze various critical phenomena for a model of long homogeneous polymer chains in an attracting potential field. The model exhibited here demonstrates a phase transition from a densely packed globular phase at low temperatures to an extended phase at higher temperatures. In the latter phase, the thermal fluctuations overcome the attraction between monomers and the chain takes on the shape of a 3d random walk or Brownian motion with a typical scale O( √ T ) where T is the length of the polymer. A real life example of this phenomenon is that of albumen (egg white). We describe a rough picture of this situation. The physical reality is more complex as there are present several types of protein with different critical points. However in a simplified version, at room temperature the albumen is in the globular state and as a result, it forms a viscous, translucent liquid. However, at higher temperatures (around 60 − 65 o C) there is a transition of the albumen to a diffusive (extended) state resulting in an opaque semi-solid material. While this transition may be reversible for an individual polymer, in the aggregate, the polymer strands in the diffusive state become interwoven and form chemical bonds with each other and can not return to the globular state when the temperature is decreased.
It is worthwhile recalling Gibbs' philosophy of phase transitions. Start with a system of finite size T . The configuration space Σ T = {x(·)} denotes all possible states x(·) of the system. The space Σ T is equipped with a reference measure P 0,T which corresponds to infinite absolute temperature (in our case, the inverse temperature β = 0). The configurations satisfy boundary conditions which reflect the interaction of the finite system with its environment. This system is endowed with a Hamiltonian H T giving the energy H T (x) of the state x. For β > 0, the Gibbs measure P β,T is given by the density where When T < ∞, the measure P β,T and the thermodynamic quantities associated to P β,T are analytic functions of β. Now let T → ∞. In typical situations, there is a critical value β cr such that for β > β cr , there exists a unique limiting measure P β on Σ, the space of infinite configurations, and this limiting measure is independent of the boundary conditions on Σ T . Moreover, P β and its relevant thermodynamic quantities are still analytic functions of β for β > β cr . One manifestation of the phase transition is the non-uniqueness for β < β cr of the limiting measure as T → ∞ as it has dependence on the boundary conditions on Σ T . Another is the non-analyticity of thermodynamic quantities associated to P β as a function of β. The mathematical characterization of the phase transition in terms of non-uniqueness of the limiting Gibbs measure traces its history to the works of Dobrushin [2] and Ruelle [7].
Modern physical theories predict that near the critical point β = β cr the limiting Gibbs measure P β must be invariant with respect to renormalizations of the system (selfsimilarity). This idea is related to the two-parametric scaling by Fisher [3] for β near β cr . Another important fact is that critical behavior as β → β cr of the physical system demonstrates universality, that is the same behavior holds for a wide class of Hamiltonians.
The most essential part of the present paper is the detailed description of the polymer chain near the critical point and the establishment of the physical ideas of universality and self-similarity for our particular model of homopolymers.

Description of the Model and Results
A continuous function x : [0, T ] → R d , x(0) = 0, will be thought of as a realization of the polymer. The parameter t ∈ [0, T ] can be intuitively understood as the length along the polymer (although the functions x = x(t) are not differentiable and the genuine notion of length can not be defined).
We assume that for β = 0, the polymer is distributed according to the Wiener measure P 0,T on Σ T = C([0, T ], R d ). For an infinitely smooth compactly supported potential v ∈ C ∞ 0 (R d ) and a coupling constant β ≥ 0, the polymer is distributed according to the Gibbs measure P β,T , whose density with respect to P 0,T is In other words, the Hamiltonian H T is given by H T = − T 0 v(x(s))ds. The normalizing factor Z β,T , called the partition function, is given by It will be usually assumed that the potential is nonnegative and not identically equal to zero. We shall be interested in the prevalent behavior of the polymer with respect to the measure P β,T as T → ∞.
We shall see that there are two qualitatively different cases corresponding to different values of β. Namely, for all sufficiently large values of β there is a limiting distribution for x(T ) with respect to P β,T . Moreover, for each positive constant s and each function S(T ) such that S(T ) → ∞ and T − S(T ) → ∞ as T → ∞, the family of processes x(S(T ) + t), t ∈ [0, s], with respect to either measure P β,T or P β,T (·|x(T ) = 0), converges to a Markov process as T → ∞. The generator of the limiting Markov process and its invariant measure are written out explicitly in Theorem 8. 3. Since x(S(T )) and x(T ) converge to limiting distributions and thus typically remain bounded as T → ∞, we shall say that the polymer is in the globular state.
We shall see that there is a number β cr (called the critical value of the coupling constant) such that the polymer is in the diffusive state for β < β cr and in the globular state for β > β cr . The value of β cr and the behavior of the polymer when β is near β cr depend on the dimension d and on the potential. In particular, we shall see that β cr = 0 for d = 1, 2 and β cr > 0 for d ≥ 3.
Of particular interest is the behavior of the polymer when β = β cr . In this case the appropriate scaling is the same as in the diffusive case, that is we study the family of processes x(tT )/ √ T , 0 ≤ t ≤ 1. We shall find the limit of this family as T → ∞. It turns out to be a Markov process with a non-Gaussian, spherically symmetric transition function (Theorem 10.6). The transition function of the limiting Markov process will be written out explicitly.
In order to determine whether the polymer is in the globular or diffusive state for a given β, we shall look at the rate of growth of the partition function Z β,T . Namely, let T .
It will be demonstrated that the limit exists and is equal to the supremum of the spectrum of the operator H β = 1 2 ∆ + βv : The infimum of the set of β for which λ 0 (β) > 0 is equal to β cr . It will be seen that λ 0 (β cr ) = 0 is an eigenvalue of H βcr in dimensions d ≥ 5, and corresponds to a ground state of H βcr in dimensions d = 3, 4.
The paper is organized as follows.
In Section 3 we consider finite T and show that {x(t), 0 ≤ t ≤ T } is a timeinhomogeneous Markov process with respect to the measures P β,T and P β,T (·|x(T ) = 0).
In Section 4 we prove the existence of the critical value of the coupling constant. In Section 5 we analyze the properties of the resolvent of the operator H β which, in particular, will be needed to study the asymptotic properties of the partition function.
In Section 6 we shall examine the asymptotics of λ 0 (β) when β ↓ β cr and show it has the following asymptotic behavior as β ↓ β cr , These asymptotics demonstrate universality in that they depend only on dimension. The constants c d , d ≥ 3, are not universal however. In Section 7 we find the asymptotics, as T → ∞, of Z β,T . In particular, when β > β cr , we shall find that Z β,T ∼ k β e λ 0 (β)T for some constant k β , while for β < β cr , Z β,T has a finite limit as T → ∞. Finally, when β = β cr , it turns out that Z β,T ∼ k 3 T 1/2 for d = 3, Z β,T ∼ k 4 T /ln T for d = 4, while Z β,T ∼ k d T for d ≥ 5. We also give asymptotics of the solutions to the parabolic equation ∂u/∂t = H β u.
In Sections 8, 9 and 10, we describe the behavior of the polymer for β > β cr , β < β cr and β = β cr , respectively, establishing the convergence results mentioned above.
Some of the results presented above have been obtained by Cranston and Molchanov in [1] for the discrete model with the potential concentrated at one point. The analysis was based on explicit formulas for the solution of the parabolic equation with such a potential. The current results demonstrate that the behavior of the polymer is "universal" with respect to the choice of the potential. Another essential feature of this paper is the detailed analysis of the behavior of the polymer when β = β cr . We refer the reader to the review of Lifschitz, Grosberg and Khokhlov [5] for a wealth of information and ideas on polymer chains.

Time-inhomogeneous Markov Property
First we define p β as the fundamental solution of the heat equation In this section we shall prove that with respect to the measure P β,T , the process {x(t), 0 ≤ t ≤ T } is a time-inhomogeneous Markov process. Since we shall point out the link between non-uniqueness of Gibbs measures and phase transitions it will be necessary to also consider the transition mechanism for the process {x(t), 0 ≤ t ≤ T } under the conditional measure P β,T (·|x(T ) = 0). Namely, we will show that the free boundary condition corresponding to the measure P β,T and the pinned boundary condition corresponding to the measure P β,T (·|x(T ) = 0) lead to different Gibbs measures in the limit.
where E x is the expectation with respect to the measure induced by the Brownian motion starting at x. Thus Z β,t (0) = Z β,t , where Z β,t is the partition function introduced in the previous section.
Theorem 3.1. The process {x(t), 0 ≤ t ≤ T } is a time-inhomogeneous Markov process with respect to the measures P β,T . Its transition density is given by The transition density q T β ((s, y), (t, x)) solves the parabolic equation ∂ ∂s q T β ((s, y), (t, x))+ 1 2 ∆ y q T β ((s, y), (t, x)) + ∇ y ln Z β,T −s (y)∇ y q T β ((s, y), (t, x)) = 0. (6) With respect to the conditional measure P β,T (· |x(T ) = 0), the process {x(t), 0 ≤ t ≤ T } is a time-inhomogeneous Markov process with transition density While this result is not used directly in later sections, it provides some intuition on the nature of the limiting processes when we consider the limit T → ∞.
Proof. The Feynman-Kac formula gives that for 0 < t ≤ T , Similarly, for 0 = t 0 < t 1 < t 2 < ... < t n ≤ T and x 0 = 0, So, if we set for 0 ≤ s < t ≤ T , Since q T β ((s, y), (t, x)) > 0 and this means that {x(t), 0 ≤ t ≤ T } under the measure P β,T is a time-inhomogeneous Markov process with transition probabilities q T . Turning the equation for q T β around and solving for p β yields Using the fact that ∂ ∂s p β (t − s, y, x) + 1 2 ∆ y p β (t − s, y, x) + βv(y)p β (t − s, y, x) = 0, we derive that q T β satisfies the equation Simplifying this leads to the following parabolic equation for q T β , ∂ ∂s q T β ((s, y), (t, x))+ 1 2 ∆ y q T β ((s, y), (t, x)) + ∇ y ln Z β,T −s (y)∇ y q T β ((s, y), (t, x)) = 0. (10) Next we consider the pinned case, for 0 = t 0 < t 1 < ... < t n < t n+1 = T and x 0 = x n+1 = 0. Then, Now set for 0 ≤ s < t ≤ T , Then Since q We shall see below in that in the globular phase β > β cr the drift term ∇ x ln Z β,T −s (x) has a non-trivial limit as T → ∞. This means that for β > β cr , the Gibbs measure corresponds to a stationary Markov process in the T → ∞ limit. On the other hand, this limit will vanish for β < β cr . This explains the nature of the diffusive state for high temperature.

Critical Value of the Coupling Constant
Let We shall always assume that v(x) is non-negative and compactly supported, although many results do not require these restrictions or can be modified to be valid without these restrictions. We shall also assume that v is not identically equal to zero. It is well-known that the spectrum of H β consists of the absolutely continuous part (−∞, 0] and at most a finite number of non-negative eigenvalues: We enumerate the eigenvalues in the decreasing order. Thus, if {λ j } = ∅, then λ 0 = max λ j . Lemma 4.1. There exists β cr ≥ 0 (which will be called the critical value of β) such that sup σ(H β ) = 0 for β ≤ β cr and sup σ(H β ) = λ 0 (β) > 0 for β > β cr . For β > β cr the eigenvalue λ 0 (β) is a strictly increasing and continuous function of β. Moreover, lim β↓βcr λ(β) = 0 and lim β↑∞ λ(β) = ∞.
Other statements easily follow from the fact that for each ψ the form (H β ψ, ψ) depends continuously and monotonically on β.
Remark. As will be shown below, β cr = 0 for d = 1, 2, and β cr ≥ 0 for d ≥ 3. Thus we do not talk about phase transition for d = 1, 2 since we do not consider negative values of β.
For d ≥ 3, by the Cwikel-Lieb-Rozenblum estimate [6], This implies that there are no eigenvalues for sufficiently small values of β if d ≥ 3, that is β cr > 0. It is also well-know (see [6]) that sup σ(H β ) > 0 for d = 1, 2 if β > 0, v ≥ 0 and v is not identically zero. These statements will also be proved below without referring to the Cwikel-Lieb-Rozenblum estimate.

Analytic Properties of the Resolvent
The resolvent of the operator H β will be considered in the spaces of square-integrable and continuous functions.
is a meromorphic operator valued function on C ′ = C\(−∞, 0]. Denote the kernel of R β (λ) by R β (λ, x, y). If β = 0, the kernel depends on the difference x − y and will be denoted by R 0 (λ, x − y). The kernel R 0 (λ, x) can be expressed through the Hankel function H ν : and In particular, We shall say that f ∈ L 2 exp (R d ) if f is measurable and Similarly, we shall say that f ∈ C exp (R d ) if f is continuous and Note that R 0 (λ), λ ∈ C ′ , is a bounded operator not only in L 2 (R d ) but also from The well-known properties of the Hankel functions together with (14) and (15) imply the following lemma (see [8] for a similar statement for general elliptic operators).

3) The operator A(λ) is compact for all
The operator A(λ) has the following asymptotic behavior as λ → 0, λ ∈ C ′ : , have the following kernels: Proof. Let d be odd. From (14), (15) and (16) it follows that the kernel A(λ, . The kernel has a weak singularity at x = y and an exponential estimate at infinity. To be more exact, were C(d, k) has a singularity at k = 0 if d = 1. Since the estimate (17) immediately leads to the analyticity in k = √ λ of the operator A(λ) in the space C exp (R d ). In order to get the same result in the space Here χ is the indicator function of the unit ball. Since the convolution with χ(x)R 0 (k 2 , x) is an analytic in k operator in the space L 2 (R d ). Then B 1 (which is the convolution followed by multiplication by v(x)) is an analytic operator in the space L 2 exp (R d ). The product of the kernel of the operator B 2 and e |x| 2 −|y| 2 is square integrable in (x, y). The same is true for the derivative in k of the kernel of B 2 multiplied by e |x| 2 −|y| 2 . Thus B 2 is also analytic in k. This completes the proof of the analyticity of A(λ) when d is odd. The case of even d is similar. One needs only to take into account that R 0 (λ, x) has a logarithmic branching point at λ = 0 in this case. The second statement of the lemma follows immediately from (14), (15) and (16).
To prove the compactness of A(λ), we note that the estimate (17) is valid not only for A(k 2 , x, y) and ∂A(k 2 , x, y)/∂k, but also for ∇ x A(k 2 , x, y). Thus the arguments above lead to the boundedness of the operators ∂ ∂x i A(λ) (the composition of A(λ) with the differentiation). Since the supports of functions A(λ)f belong to the support of v, the standard Sobolev embedding theorems imply the compactness of the operator A(λ) in both the spaces L 2 exp (R d ) and C exp (R d ). In order to prove the fourth statement of the lemma, we observe that the L 2 (R d ) norm of the resolvent R 0 (λ) does not exceed 1/|Imλ| (the inverse distance from the spectrum). Since A(λ) is obtained from R 0 (λ) after multiplying it by a bounded function with compact support, the L 2 exp (R d ) norm of A(λ) does not exceed c/|Imλ|, where c is a positive constant which depends on v. The norm of A(λ) in the space C exp (R d ) can be estimated by The remaining statements also easily follow from (14) and (15). Note that for d ≥ 3, there exists the limit which is a fundamental solution of the operator 1 2 ∆. The operator with this kernel will be denoted by The following lemma follows from formulas (14) and (15) similarly to Lemma 5.1.
It has the following asymptotic behavior as λ → 0, The following lemma is simply a resolvent identity. It plays an important role in our future analysis. Lemma 5.3. For λ ∈ C ′ , we have the following relation between the meromorphic operator-valued functions Remark. Note that (19) can be written as From here it also follows that which should be understood as an identity between meromorphic in λ operators acting In the lattice case considered in [1], the operator A(λ) has rank one and where I(λ) is an analytic function of √ λ related to A(λ). This exact formula is the key to all the results in [1].
The kernels of the operators I + βA(λ) (both in spaces L 2 exp (R d ) and C exp (R d )) are described by the following lemma. (2) Let λ i (β) be a positive eigenvalue of H β . There is a one-to-one correspondence between the kernel of the operator I + βA(λ i ) and the eigenspace of the operator H β corresponding to the eigenvalue λ i . Namely, if there is a one-to-one correspondence between the kernel of the operator I + βA(0) and solution space of the problem Remark. The relations (22) are an analogue of the eigenvalue problem for zero eigenvalue and the eigenfunction ψ which does not necessarily belong to L 2 (R d ) (see Lemma 5.6 below). We shall call a non-zero solution of (22) a ground state.
If λ ∈ C ′ is a pole of (I + βA(λ)) −1 , then it is also a pole of the same order of R β (λ) as follows from (21) since the kernel of R 0 (λ) is trivial. Therefore the pole is simple and coincides with one of the eigenvalues λ i . Note that λ is a pole of (I + βA(λ)) −1 if and only if the kernel of I + βA(λ) is non-trivial. Let h ∈ L 2 exp (R d ) be such that ||h|| L 2 exp (R d ) = 0 and (I + βvR 0 (λ))h = 0. Then ψ : Conversely, let ψ ∈ L 2 (R d ) be an eigenfunction corresponding to an eigenvalue λ i , that is ( , and therefore belongs to the kernel of I + βA(λ i ). This completes the proof of the first two statements.
Similar arguments can be used to prove the last statement. If h ∈ L 2 exp (R d ) is such that ||h|| L 2 exp (R d ) = 0 and (I + βA(0))h = 0, then h has compact support and the integral operator R 0 (0) can be applied to h. It is clear that ψ := −R 0 (0)h satisfies (22) and, since h has compact support, h ∈ C exp (R d ).
In order to prove that any solution of (22) corresponds to an eigenvector of I + βA(0), one only needs to show that the solution ψ of the problem (22) can be represented in the form ψ = −R 0 (0)h with h = βvψ. The latter follows from the Green formula after passing to the limit as a → ∞. Lemma 5.4 can be improved for λ = λ 0 (β). Due to the monotonicity and continuity of λ = λ 0 (β) for β > β cr , we can define the inverse function We shall prove that the operator −A(λ), λ > 0, has a non-negative kernel and has a positive simple eigenvalue such that all the other eigenvalues are smaller in absolute value. Such an eigenvalue is called the principal eigenvalue.
has the principal eigenvalue. This eigenvalue is equal to 1/β(λ) and the corresponding eigenfunction can be taken to be positive in the interior of supp(v) and equal to zero outside of supp(v). If d ≥ 3, then the same is true for the operator −A(0) (in particular, β cr > 0). Remark 1. Let d ≥ 3. Lemmas 5.4 and 5.5 imply that the ground state of the operator H β for β = β cr (defined by (22)) is defined uniquely up to a multiplicative constant and corresponds to the principal eigenvalue of A(0). The ground state (with λ = 0) does not exist if β < β cr .
Therefore for all λ ∈ C ′ with |λ| sufficiently small, the operator −A(λ) has a simple eigenvalue whose real part is larger than the absolute values of the other eigenvalues. We shall denote this eigenvalue by 1/β(λ), thus extending the domain of the function β(λ) (see Proof of Lemma 5.5. By Lemma 5.4 it is sufficient to consider the case of L 2 exp (R d ). The maximum principle for the operator ( 1 2 ∆ − λ), λ > 0, implies that the kernel of the operator R 0 (λ), λ > 0, is negative. Thus, by (16), for all y the kernel of −A(λ) is positive when x is in the interior of supp(v) and zero otherwise. Thus −A(λ), λ > 0, has the principal eigenvalue (see [4]). On the other hand, by Lemma 5.4, 1/β(λ) is a positive eigenvalue of −A(λ). Note that this is the largest positive eigenvalue of −A(λ). Indeed, if µ = 1/β ′ > 1/β(λ) is an eigenvalue of −A(λ), then λ is one of the eigenvalues λ i of H β ′ by Lemma 5.4. Therefore, λ i (β ′ ) = λ 0 (β) for β ′ < β. This contradicts the monotonicity of λ 0 (β). Hence the statement of the lemma concerning the case λ > 0 holds.
The relationship between ground states and eigenfunctions of H β is explained by the following lemma. Proof. The ground states belong to L 2 (R d ) if and only if d ≥ 5. In order to complete the proof of the lemma, it remains to show that any eigenfunction of H β with zero eigenvalue satisfies (22). Thus, it is enough to prove that if 1 2 The latter follows from the dominated convergence theorem.
The following lemma summarizes some facts about the operator (I + βA(λ)) −1 proved above. It also describes the structure of the singularity of the operator (I + βA(λ)) −1 for λ and β in a neighborhood of λ = 0, β = β cr .
For d ≥ 3, let h λ be an eigenvector corresponding to the eigenvalue 1/β(λ) of the operator −A(λ), λ ∈ [0, ∞) ∪ (U ∩ C ′ ). By Lemma 5.5 and the second remark following it, this eigenvector is defined up to a multiplicative constant. Let A * (λ) be the opera- Similarly to Lemma 5.5, it is not difficult to show that 1/β(λ) is an eigenvalue for the operator −A * (λ) and that its real part exceeds the absolute values of the other eigenvalues. The corresponding eigenvector h * λ is uniquely defined up to a multiplicative constant. Moreover, we can take h λ and h * λ such that Note that h λ and h * λ can be chosen in such a way that for some k > 0 and all sufficiently small |λ|, where the norms on both sides of (29) are either in the space L 2 exp (R d ) or C exp (R d ). Recall that A(λ) → A(0) as λ → 0, λ ∈ C ′ , by Lemma 5.1. Using this and the fact that 1/β cr is the principal eigenvalue for −A(0), it is easy to show that there are λ 1 > 0 and δ 1 > 0 such that for λ ∈ C ′ ∪ {0}, |λ| ≤ λ 1 , the eigenvalue 1/β(λ) of the operator −A(λ) is the unique eigenvalue whose distance from 1/β cr does not exceed δ 1 . Take 0 < λ 0 < λ 1 and 0 < δ 0 < δ 1 such that for λ ∈ C ′ ∪ {0}, |λ| ≤ λ 0 , the distance between 1/β(λ) and 1/β cr does not exceed δ 0 .
Then for λ ∈ C ′ ∪ {0}, |λ| ≤ λ 0 and β such that |1/β − 1/β cr | ≤ δ 0 , the operator valued function is meromorphic inside the circle γ = {z : |z−1/β cr | = δ 1 }. It has two poles: one at z = 1/β and the other at z = 1/β(λ). The residue at the first pole is equal to (A(λ) + I/β) −1 . In order to find the residue at the second pole, recall that it is a simple pole for (A(λ)+zI) −1 , and therefore for some operators T −1 , T 0 , T 1 , ... and all z in a neighborhood of 1/β(λ). From here and the fact that the kernels of A(λ) + I/β(λ) and A * (λ) + I/β(λ) are one-dimensional and coincide with span{h λ } and span{h * λ }, respectively, it easily follows that From (29) and Lemma 5.1 it follows that S d (λ) : Integrating F (z) over the contour γ, we obtain The right hand side of this formula is uniformly bounded, which completes the proof of the lemma if we show that T −1 (0) = B. Thus it remains to prove that The latter follows from the relation h 0 = βvψ (see Lemma 5.4) and (28). Formula (21) and Lemmas 5.2 and 5.7 imply the following result.

The Behavior of the Principal Eigenvalue for β ↓ β cr
In Lemma 5.5 we showed that β cr > 0 for d ≥ 3. The following theorem implies, in particular, that β cr = 0 for d = 1 or 2.
Proof. Since we are interested in the behavior of λ 0 (β) for β ↓ β cr and λ 0 (β) ↓ 0 when β ↓ β cr by Lemma 4.1, we shall study the behavior of β(λ) as λ ↓ 0 (or, more generally, as λ → 0, λ ∈ C ′ ). The arguments below are based on Lemma 5.1. First consider the case d = 1. For λ → 0, λ ∈ C ′ , the eigenvalue problem for −A(λ) can be written in the form Note that the kernel of vP 1 is positive when x is an interior point of supp(v). Therefore vP 1 has a principal eigenvalue. In fact, the operator vP 1 is one-dimensional and the eigenvalue is equal to Since this eigenvalue is simple and the operator in the left-hand side of (36) is analytic in √ λ, both h λ and √ λ/β(λ) are analytic functions of √ λ in a neighborhood of the origin and lim λ→0,λ∈C ′ ( Therefore, β cr = 0, β(λ) is analytic in √ λ, and β(λ) ∼ √ 2λ/c 1 as λ → 0, λ ∈ C ′ , which proves (31).
The same arguments in the case d = 2 lead to the relation This implies that β cr = 0 and (32) holds.
In the case d = 3 the eigenvalue problem for −A(λ) takes the form As in the one-dimensional case, 1/β(λ) and h λ are analytic functions of √ λ. Now 1/β cr is equal to the principal eigenvalue of −A(0). Recall that h 0 is the principal eigenfunction of −A(0) and h * 0 is the principal eigenfunction of −A * (0). Standard perturbation arguments imply that 1 where which implies (33) with c 3 = 1/(γ 2 β 4 cr ). Note that γ > 0 since the kernel of the operator vQ 3 is negative and principal eigenfunctions h 0 , h * 0 can be chosen to be positive inside supp(v).
Formula for γ can be simplified. We choose h 0 = βvψ (see Lemma 5.4) and h * 0 defined in (28). Then Let d = 4. Then instead of (37) we get From here it follows that where 1/β cr is the principal eigenvalue of −A(0) and γ is given by (39) with Q 3 replaced by Q 4 . Thus (34) holds with c 4 = 1/(γβ 2 cr ). For d ≥ 5 we get From here it follows that where 1/β cr is the principal eigenvalue of −A(0) and γ is given by (39) with Q 3 replaced by Q d . Thus (35) holds with c d = 1/(γβ 2 cr ).

Asymptotics of the Partition Function, Solutions, and Fundamental Solutions
We shall need the following notation. Recall from (4) that by p β (t, y, x) we denote the fundamental solution of the parabolic problem .
For a given f ∈ L 2 (R d ), let be the solution of the Cauchy problem with the initial data f . The partition function is defined as the integral of the fundamental solution Note that the partition function defined in (3) is simply Z β,T = Z β,T (0). Also note that Z β,t (x) is the solution of the Cauchy problem with initial data equal to one: For β > β cr , let ψ β be the positive eigenfunction for the operator H β with eigenvalue λ 0 (β) normalized by the condition ||ψ β || L 2 (R) = 1. This function is defined uniquely by Lemma 5.4 and is equal to −R 0 (λ)h λ , where λ = λ 0 (β) and h λ is the principal eigenfunction for the operator −A(λ). Note that ψ β decays exponentially at infinity.
For a ∈ R, let Γ(a) be the following contour in the complex plane We choose the direction along Γ(a) in such a way that the imaginary coordinate increases. The following lemma is an important tool for investigating the asymptotics of Z β,T .
which holds in L 2 (R d ) (or C(R d )). This formula remains valid if the initial function f is identically equal to one and R β (λ)f is understood by substituting f ≡ 1 into (19) with R 0 (λ)1 = −1/λ. More precisely, in L 2 (R d ) and C(R d ).
Proof. First, let f ∈ L 2 (R d ). We solve the Cauchy problem for u β using the Laplace transform with respect to t. This leads to (43) with Γ(a) replaced by the line {λ : Reλ = a}. The integral over this line is equal to the integral over Γ(a) since the resolvent is analytic between these contours and its norm decays as |λ| −1 when |λ| → ∞. Now let f ≡ 1. Then w(t, x) = Z β,t (x) − 1 is the solution of the problem By the Duhamel formula and (43), since in the domain Γ + (a) to the right of the contour Γ(a), the operator R β (λ) : is analytic and decays as |λ| −1 at infinity. This justifies (44) in L 2 (R d ) sense. It remains to show that the right-hand side of (43) is continuous for f ∈ C exp (R d ) and the right-hand side of (44) is continuous. Since βv ∈ C ∞ 0 , the integrands are continuous in (t, x) for each λ ∈ Γ a . It remains to note that the integrals converge uniformly when x ∈ R n , t ≥ t 0 > 0. This is due to the fact that ||R β (λ)f || C(R d ) , ||R β (λ)βv|| C(R d ) ≤ C d (a), as follows from Lemma 5.8.
In order to state the next theorem we shall need the following notation. As in part (3) of Lemma 5.4, it is not difficult to show that for d ≥ 3, 0 ≤ β < β cr and f ∈ C ∞ 0 (R d ) there is a unique solution of the problem (45) This solution is given by ϕ = R 0 (0)(I + βA(0)) −1 f . For f = −βv, we denote this solution by ϕ β .
(1) For β > β cr there is ε > 0 such that we have the following asymptotics for the partition function: which holds in L 2 (R d ) and in C(R d ), where ψ β is the positive eigenfunction for the operator H β with eigenvalue λ 0 (β) normalized by the condition ||ψ β || L 2 (R) = 1.
(2) For β = β cr we have the following asymptotics for the partition function: which holds in C(R d ). Here k d , d ≥ 3, are positive constants and ψ is the positive ground state for H βcr normalized by the condition ||β cr vψ|| L 2 exp (R d ) = 1. Proof.
It remains to show that the second term on the right-hand side of (46) is exponentially smaller than the first term. This is due to the fact that the norm of the operator R β (λ) is of order 1/|λ| at infinity for λ ∈ Γ(λ 0 (β) − ε).
We choose the same ground state ψ specified in the statement of Theorem 7.2. Then from (26) and Lemma 5.4 it follows that Now, by Lemma 5.8 and (33), (38), where the remainder D(λ) is of order O(1) when λ → 0, λ ∈ C ′ . Note that D(λ) is bounded on Γ + (0) since the left hand side and the first term on the right hand side of (48) are bounded on Γ + (0) outside a neighborhood of zero. Next, we apply (44) with a replaced by 1/t and use the expression (48) to obtain Let us change the variables in the integral λt = z. Thus The contribution to the integral from the term containing D(z/t) is bounded, while the contribution from the first term is equal to k 3 t 1/2 ψ(x), as claimed in the lemma. One needs only to note that k 3 > 0 since If d = 4, then (34), (42) imply that β(λ) − β cr ∼ β 2 cr γλ ln(1/λ) as λ → 0, λ ∈ C ′ . This leads to the following analog of (49) where D(λ) is of order O(1/|λ ln 2 λ|) when λ → 0, λ ∈ C ′ and is bounded at infinity. After the change of variables λt = z, we obtain which easily leads to the second part of the lemma in the case d = 4. The treatment of the case d ≥ 5 is similar.
(3) We apply (44) with a replaced by 1/t to obtain Note that by Lemma 5.2 and since 1/β is not an eigenvalue of A(0) we have Since the difference between R β (z/t)(βv) and −ϕ β is bounded on Γ(1), one can pass to the limit t → ∞ under the integral sign in (50), which leads to The third part of Theorem 7.2 establishes the existence of lim t→∞ Z β,t (x) for β < β cr . Next we examine the behavior of this quantity as β ↑ β cr .
where ψ is the positive ground state for H βcr normalized by the condition ||β cr vψ|| L 2 exp (R d ) = 1.

Behavior of the Polymer for β > β cr
In this section we shall assume that β > β cr is fixed. A result similar to the first part of Theorem 7.2 is valid for the solution of the Cauchy problem and for the fundamental solution.
. For β > β cr there is ε > 0 such that we have the following asymptotics for the solution u β of the Cauchy problem with the initial data f : which holds in L 2 (R d ) (or in C(R d )), where ||q f (t)|| ≤ c||f || exp(−εt) for some c and all sufficiently large t.
We have the following asymptotics for the fundamental solution of the parabolic equation: where lim t→∞ ||q(t, y, x)|| = 0, uniformly in y, and (52) holds in L 2 (R d ) and in C(R d ) for each y fixed.
Proof. The proof of (51) is the same as the proof of the first part of Theorem 7.2, and therefore we omit it. Let f δ,y β (x) = p β (δ, y, x) be the fundamental solution of the parabolic problem at time δ. Note that f δ,y β ∈ L 2 (R d ) for all δ > 0 and all y, and f δ,y β ∈ C exp (R d ) for all sufficiently small δ > 0 and all y. Denote the solution of the parabolic equation with the initial data f δ,y β by u δ,y β (t, x). Then where ||q δ (t, y, x)|| ≤ c||f δ,y β || exp(−ε(t − δ)) for some c and all sufficiently large t. Note that ψ β , f δ,y β L 2 (R d ) can be made arbitrarily close to ψ β (y) uniformly in y, by choosing a sufficiently small δ, and ||f δ,y β || is uniformly bounded in y for any fixed δ . This justifies (52).
Next, let us study the distribution of the end of the polymer with respect to the measure P β,T as T → ∞.
Theorem 8.2. The distribution of x(T ) with respect to the measure P β,T converges, weakly, as T → ∞, to the distribution with the density ψ β /||ψ β || L 1 (R d ) .
Proof. The density of x(T ) with respect to the Lebesgue measure is equal to where q is the same as in (52). When T → ∞, the right hand side of (53) converges to Note that r β (t, y, x) is the fundamental solution for the operator ∂/∂t − L * β , where L * β is the formal adjoint to L β . Thus r β is the transition density for the Markov process with the generator L β . Also note that L * β ψ 2 β = 0, and thus ψ 2 β is the invariant density for the Markov process.
Proof of Theorem 8.3. We shall only consider the measure P β,T since the arguments for the measure P β,T (·|x(T ) = 0) are completely analogous. First, let us prove the convergence of the finite-dimensional distributions. For y ∈ R d and a Borel set A ∈ B(R d ), let R(t, y, A) = A r β (t, y, x)dx, with r β given by (55). Note that R is a Markov transition function since The generator of the corresponding Markov process is L β and the invariant density is ψ 2 β . Let 0 ≤ t 1 < ... < t n ≤ s. The density of the random vector (y T (t 1 ), ..., y T (t n )) with respect to the Lebesgue measure on R dn is equal to We replace here all factors p β , except the first one, by r β using (55). We replace the first factor and the factors Z by their asymptotic expansions given in Theorems 8.1 and 7.2, respectively. This leads to where the remainder tends to zero uniformly in (x 1 , ..., x n ). By the remark made after the statement of the theorem, this justifies the convergence of the finite dimensional distributions of y T to those of the Markov process. It remains to justify the tightness of the family of measures induced by the processes y T . From the convergence of the one-dimensional distributions it follows that for any η > 0 there is a > 0 such that P β,T (|y T (0)| > a) ≤ η.
Let us prove that for each ε, η > 0, there is δ > 0 such that for all sufficiently large T . Observe that where C(δ, ε) is the probability that for a d-dimensional Brownian motion W t , 0 ≤ t ≤ s, we have sup Note that is bounded, as follows from Theorems 7.2 and 8.1, while C(δ, ε) can be made arbitrarily small by selecting a sufficiently small δ. This justifies (57). Since the inequalities (56) and (57) hold for all sufficiently large T , by choosing different a and δ, we can make sure that they hold for all T . Thus the family of measures induced by the processes y T is tight.
Remark. If instead of (54) we assume that S(T ) = 0, the result of Theorem 8.3 will hold with the only difference that the initial distribution for the limiting Markov process will now be concentrated at zero, instead of being the invariant distribution.
9 Behavior of the Polymer for β < β cr First, we shall study the asymptotic behavior of the solution u β (t, x) of the Cauchy problem and of the fundamental solution p β (t, y, x) when t → ∞, |y| ≤ ε −1 , ε √ t ≤ |x| ≤ ε −1 √ t, and ε > 0 is small but fixed. Recall that ϕ β was defined before Theorem 7.2.
Lemma 9.1. Let d ≥ 3, 0 ≤ β < β cr , ε > 0 and f ∈ C exp (R d ), f ≥ 0. We have the following asymptotics for the solution u β of the Cauchy problem with the initial data f : where for some constant C β (ε) we have We have the following asymptotics for the fundamental solution of the parabolic equation: where lim Proof. Note that (59) follows from (58) since the fundamental solution at time t is equal to the solution with the initial data p β (t, y, δ) evaluated at time t − δ (the same argument was used in the proof of Theorem 8.1). Therefore it is sufficient to prove (58). For the sake of transparency of exposition, we shall consider only the case d = 3. From Lemma 5.8 it follows that we can put a = 0 in (43) when β < β cr . Thus using (21) and the explicit formula for R 0 (λ), we obtain where g(λ) = (I + βA(λ)) −1 f.
Fix ε > 0. For a continuous function x : Let E δ be the event that m(x, δ) > ε/4 and E δ the event that m(x, δ, ε/4) > ε/4. For 0 ≤ s ≤ T , let E s δ be the event that a continuous function x : where E x 0,T denotes the expectation with respect to the measure induced by the Brownian motion starting at the point x. Since it is sufficient to estimate βv(x(t))dt)).
Let E ′ be the event that a trajectory starting at x reaches the support of v before time T . Note that lim The expression in (69) is estimated form above by βv(x(t))dt)) + P x 0,T (E δ )).
The second term does not depend on T due to the scaling invariance of the Brownian motion, and can be made arbitrarily small by selecting a sufficiently small δ. Due to the Markov property of the Brownian motion, the first term is estimated from above by and thus tends to zero when T → ∞.
10 Behavior of the Polymer for β = β cr In this section we assume that d = 3. Again, we start with the asymptotic behavior of the solution u β (t, x) of the Cauchy problem and of the fundamental solution p β (t, y, x) when t → ∞, |y| ≤ ε −1 , ε √ t ≤ |x| ≤ ε −1 √ t, and ε > 0 is small but fixed. Recall that ψ is the positive ground state for H βcr normalized by the condition ||β cr vψ|| L 2 exp (R 3 ) = 1 (see the remark following Lemma 5.4 and Theorem 7.2). For f ∈ C exp (R 3 ), define We can formally apply this to f being the δ-function centered at a point y, and thus define α(δ y (x)) = κψ(y).
Theorem 10.1. Let d = 3, β = β cr , ε > 0 and f ∈ C exp (R 3 ), f ≥ 0. We have the following asymptotics for the solution u β of the Cauchy problem with the initial data f : where for some constant C β (ε) we have We have the following asymptotics for the fundamental solution of the parabolic equation: where lim Proof. As in Lemma 9.1, formula (71) follows from (70). Lemma 5.7 implies where B is the one dimensional operator with the kernel From here, (33) and (38) we get where γ is defined in (39), (40). Hence, for any f ∈ C exp (R 3 ) and λ → 0, λ ∈ C ′ , where g 1 (λ) ≤ c||f || Cexp(R 3 ) for some constant c. Now, similarly to (60), we have The integral with g 1 (λ) instead of h can be estimated similarly to the estimate on u (1) β in the case of β < β cr . This leads to following analogue of (63) where the remainder r 1 (t, x) satisfies We evaluate the integral over Γ(0): This equality simply means that the inverse Laplace transform of the Green function of the one dimensional Helmholtz equation coincides with the fundamental solution of the corresponding heat equation. Thus, 2t v(y)ψ(y)dy + r 1 (t, x).
This implies (70) since v has a compact support. The next theorem concerns the fundamental solution when both y and x are at a distance of order √ t away from the origin. Note that now there are two terms in the asymptotic expansion for the fundamental solution which are of the same order in t. The main terms have the order t −3/2 when t → ∞, compared with t −1 in the case considered in Theorem 10.1 (where y was bounded).
In order to find the limit of the finite dimensional distributions of y T , we need to identify the limit of Q T as T → ∞. For 0 ≤ s < t ≤ 1, y ∈ R 3 and x ∈ R 3 \ {0}, define Q(s, t, y, x) = lim  p β (s, 1, y, z)dz) −1 , t < 1, Q(s, 1, y, x) = p β (s, 1, y, x)( We additionally define Q(s, t, y, 0) = 0. Using (79), (80) and (81), we can identify the limit of the densities ρ T (x 1 , ..., x n ) for x 2 , ..., x n = 0. In order to identify the weak limit of the finite dimensional distributions of the processes y T , we are going to show that the limit of the densities is the density of a probability distribution, i.e. the mass does not escape to the origin or infinity. This is done in Lemma 10.4, where we show that Q serves as the transition density for a Markov process. First, however, we show that Q satisfies a Fokker-Plank type equation on R 3 \ {0}.
By (84), For any two functions A and B we have Since g(t, x) = ln(1 + v 2 ) and 2∂v 2 /∂t = −∂ 2 v 2 /∂r 2 − 2∂v 2 /∂r (see (84)), it is easy to check that the operator in the left hand side of the equation for u is ∂ ∂t − L * , and this justifies (83).
Proof. The convergence of the finite dimensional distributions of y T (t) to those of the Markov process follows from (79) and Lemma 10.4. Since the family y T (t) is tight, there is a modification of the Markov process which has continuous trajectories.
Proof of Lemma 10.5. To prove tightness it is enough to demonstrate that for each η, ε > 0 there are 0 < δ < 1 and T 0 ≥ 1 such that for all u ∈ [0, 1] we have for some constant c(ε) for all sufficiently large T . It then remains to choose δ such that E x 0,T (χ E δ ) ≤ δη/c(ε), and the estimate (95) will follow. The second inequality in (96) easily follows from part (2) of Theorem 7.2 and the fact that the probability of reaching the support of v before time T by a Brownian path starting at a distance ε √ T /8 away from the origin is of order O(T −1/2 ) if d = 3.