ON A FUNCTION MODELING AN L-STEP SELF AVOIDING WALK

We introduce and study the needle function (Γ~a1 ◦ Vm) ◦ · · · ◦ (Γ~a l 2 ◦ Vm) : R −→ R. By exploiting the geometry of compression, we prove that this function is a function modeling an l-step self avoiding walk for l ∈ N. We show that the total length of the l-step self-avoiding walk modeled by this function is of the order l 2 √ n ( max{sup(xjk )}1≤j≤ l 2 1≤k≤n + max{sup(ajk )}1≤j≤ l 2 1≤k≤n ) and at least l 2 √ n ( min{Inf(xjk )}1≤j≤ l 2 1≤k≤n + min{Inf(ajk )}1≤j≤ l 2 1≤k≤n ) .


Introduction
Self avoiding walk, roughly speaking, is a sequence of moves on the lattice that does not visit the same point more than once. It is somewhat akin to the graph theoretic notion of a path. It is a mathematical problem to determine a function that models self avoiding walks of any given number of steps. More formally, the problem states Problem 1.1. Does there exist a function that models l-steps self-avoiding walks?
The problem had long been studied from mathematical perspective but unfortunately our understanding was not good enough. For instance the problem has recently been studied from the standpoint of network theory [2]. The problem also has great significance that extends beyond the shores of mathematics and its allied areas. For instance flurry of studies show that a good understanding of the underlying problem will certainly have its place in physics and chemistry about the long-term structural movement of substances such as polymers and certain proteins in the human anatomy [1], [3]. In this paper we find a function that models an n-step self avoiding walk. We leverage the method of compression and its accompanied estimates to study these things in much more detail. In particular we obtain the following result is the k-fold needle function with mixed translation factors a 1 , . . . , a k ∈ R n , is a function modeling l-step self avoiding walk.
We also comment very roughly about the total length of the l-step self avoiding walk modeled by the needle function in the following result

Preliminary results
In this section we recall the notion of compression and its various statistics. We find this method very efficient and much more convenient in establishing the main result of this paper. It is important to notice a compression of scale m ≥ 1 with V m : R n −→ R n is a bijective map. In particular the compression V m : R n −→ R n is a bijective map of order 2. To see why this is the case, let us suppose V m [(x 1 , x 2 , . . . , x n )] = V m [(y 1 , y 2 , . . . , y n )], then it follows that It follows that x i = y i for each i = 1, 2, . . . , n. Surjectivity follows by definition of the map. Thus the map is bijective. The latter claim follows by noting that The notion of compression is in some way the process of re scaling points in R n for n ≥ 2. Thus it is important to notice that a compression pushes points very close to the origin -with each coordinate smaller than a unit -away from the origin by certain scale and similarly draws points away from the originwith each coordinate bigger than a unit -close to the origin.
and the upper estimate follows by the estimate for this sum. The lower estimate also follows by noting the lower bound Definition 2.4. Let (x 1 , x 2 , . . . , x n ) ∈ R n with x i = 0 for all i = 1, 2 . . . , n. Then by the gap of compression, denoted G •V m [(x 1 , x 2 , . . . , x n )], we mean the expression where || x|| is the euclidean norm of the vector x = (x 1 , x 2 , . . . , x n ) or the distance of a point x = (x 1 , x 2 , . . . , x n ) relative to the origin in any euclidean space R n for any n ≥ 2.
Proposition 2.2. Let (x 1 , x 2 , . . . , x n ) ∈ R n for n ≥ 2 with x j = 0 for j = 1, . . . , n, then we have In particular for all (x 1 , x 2 , . . . , x n ) ∈ R n with x i > 1 for each 1 ≤ i ≤ n, we have the estimate Proposition 2.2 offers us an extremely useful identity. It allows us to pass from the gap of compression on points to the relative distance to the origin. It tells us that points under compression with a large gap must be far away from the origin than points with a relatively smaller gap under compression. That is to say, the inequality holds if and only if || x|| ≤ || y|| for x, y ∈ R n with x i , y j > 1 for each 1 ≤ i, j ≤ n. This important transference principle will be mostly put to use in obtaining our results.
Proof. The estimates follows by leveraging the estimates in Proposition 2.1 and noting that

Compression lines
In this section we study the notion of lines induced under compression of a given scale and the associated geometry. We first launch the following language.
Then by the line L x,Vm[ x] produced under compression V m : R n −→ R n we mean the line joining the points x and V m [ x] given by where λ ∈ R.
Remark 3.2. In striving for the simplest possible notation and to save enough work space, we will choose instead to write the line produced under compression V m : . Next we show that the lines produced under compression of two distinct points not on the same line of compression cannot intersect at the corresponding points and their images under compression. Lemma 3.3. Let a = (a 1 , a 2 , . . . , a n ), x = (x 1 , x 2 , . . . , x n ) ∈ R n with a = x and a i , x j > 1 for all 1 ≤ i, j ≤ n. If the point a lies on the corresponding line L Vm[ x] , then V m [ a] also lies on the same line.
Proof. Pick arbitrarily a point a = (a 1 , a 2 , . . . , a n ) with a i > 1 for each 1 ≤ i ≤ n and close to the point By repeating this argument under the underlying contrary assumption, we obtain an infinite descending sequence of compression gaps -lengths of distinct lines This proves the Lemma.
It is important to point out that Lemma 3.3 is the ultimate tool we need to show that certain function is indeed a function modeling l-step self avoiding walk. We first launch such a function as an outgrowth of the notion of compression. Before that we launch our second Lemma. One could think of this result as an extension of Lemma 3.3.
Lemma 3.4. Let a = (a 1 , a 2 , . . . , a n ) ∈ R n and b = (b 1 , b 2 , . . . , b n ) ∈ R n be points with identical configurations with a = b and a i , b j > 0 for 1 ≤ i, j ≤ n. If the corresponding lines L Vm[ a] : Proof. First consider the points a = (a 1 , a 2 , . . . , a n ) ∈ R n and b ) and the result follows immediately. Lemma 3.3 combined with Lemma 3.4 tells us that the line produced by compression on points with certain configuration away from other lines of compression are not intersecting. We leverage this principle to show that a certain function indeed models a self-avoiding walk.

It follows that either the point
. Under the underlying assumption, the following equations hold This is absurd since the lines L Vm[ x] and L Vm[ y] are distinct.

The ball induced by compression
In this section we introduce the notion of the ball induced by a point x = (x 1 , x 2 , . . . , x n ) ∈ R n under compression of a given scale. We study the geometry of the ball induced under compression. We launch more formally the following language.  Remark 4.2. Next we prove that smaller balls induced by points should essentially be covered by the bigger balls in which they are embedded. We state and prove this statement in the following result.
For simplicity we will on occasion choose to write the ball induced by the point x = (x 1 , x 2 , . . . , x n ) under compression as . We adopt this notation to save enough work space in many circumstances.
Theorem 4.3. Let y = (y 1 , y 2 , . . . , y n ), z = (z 1 , z 2 , . . . , z n ) ∈ R n with z i > 1 and [ y] for z = (z 1 , z 2 , . . . , z n ) ∈ R n with z i > 1 for all 1 ≤ i ≤ n, then it follows that || y|| > || z||. Suppose on the contrary that then it follows from Proposition 2.2 that || y|| < || z||, which is a contradiction. Conversely, suppose then it follows from Proposition 2.2 that || z|| ≤ || y||. It follows that This certainly implies z ∈ B 1 2 G•Vm [ y] [ y] and the proof of the theorem is complete. 4.1. Remark. As an alternative to the requirement that x ∈ R n with x i = 0 for all 1 ≤ i ≤ n, we will work under the more convenient regime that each coordinate of the vector x i > 1 for each 1 ≤ i ≤ n. This requirement has no restriction on the geometry since for a vector x = (x 1 , x 2 , . . . , x n ) with x i = 1 for each 1 ≤ i ≤ n the coordinates of the corresponding compression vector is still V 1 [ x] = (1, 1, . . . , 1). It is also easy to see that for those points x = (x 1 , x 2 , . . . , x n ) with 0 < x i < 1 the corresponding compression vector will be a point y = (y 1 , y 2 , . . . , y n ) with y i > 1 for each 1 ≤ i ≤ n and vice-versa. Thus in the sequel we will only use points x = (x 1 , x 2 , . . . , x n ) ∈ R n with x i > 1 for all 1 ≤ i ≤ n for our constructions, since the remaining points with unlike properties will automatically be obtained as compression images. It has to be said that this requirement will turn to be natural in our studies in the sequel.

It follows that
which is absurd, thereby ending the proof.

Interior points and the limit points of balls induced under compression.
In this section we launch the notion of an interior and the limit point of balls induced under compression. We study this notion in depth and explore some connections.
Definition 4.6. Let y = (y 1 , y 2 , . . . , y n ) ∈ R n with y i = 0 for all 1 ≤ i ≤ n. Then a point z ∈ B [ y] Remark 4.7. Next we prove that there must exist an interior and limit point in any ball induced by points under compression of any scale in any dimension.
contains an interior point and a limit point.
Then by employing Theorem 4.4 and Theorem 4.3, we have . By continuing the argument in this manner, we obtain the infinite descending sequence of the gap of compression thereby ending the proof of the theorem. Now we state and prove a result that in some way makes our earlier imposition in Remark 4.1 a natural one and gives more meaning to our work in further sequel.
Proof. Applying the compression V 1 : R n −→ R n on the point x = (x 1 , x 2 , . . . , x n ) with x i = 1 for each 1 ≤ i ≤ n, we obtain V 1 [ x] = (1, 1, . . . , 1)  for any y = (y 1 , y 2 , . . . , y n ) ∈ R n with y i > 1 for all 1 ≤ i ≤ n. For if the contrary [ y] holds for some y = (y 1 , y 2 , . . . , y n ) ∈ R n with y i > 1 for each 1 ≤ i ≤ n, then there must exists some point . Appealing to Theorem 4.3, we have the corresponding inequality of compression gaps so that by appealing to Proposition 2.2 and the ensuing remarks, we have the inequality of their corresponding distance relative to the origin This is a contradiction, since by our earlier assumption y = (y 1 , y 2 , . . . , y n ) ∈ R n with y i > 1 for each 1 ≤ i ≤ n. Thus the point x = (x 1 , x 2 , . . . , x n ) with x i = 1 for each 1 ≤ i ≤ n must be the limit point of any ball of the form [ y] for any y = (y 1 , y 2 , . . . , y n ) ∈ R n with y i > 1 for each 1 ≤ i ≤ n.

Admissible points of balls induced under compression.
We launch the notion of admissible points of balls induced by points under compression. We study this notion in depth and explore some possible connections.
Definition 4.9. Let y = (y 1 , y 2 , . . . , y n ) ∈ R n with y i = 0 for all 1 ≤ i ≤ n. Then y is said to be an admissible point of the ball Remark 4.10. It is important to notice that the notion of admissible points of balls induced by points under compression encompasses points on the ball. These points in geometrical terms basically sit on the outer of the induced ball. Next we show that all balls can in principle be generated by their admissible points.
Theorem 4.11. The point y Proof. First let y ∈ B By leveraging Proposition 2.2, it follows that || x|| < || y|| or || y|| < || x||. By joining this points to the origin by a straight line, this contradicts the fact that the point by Theorem 4.3. Thus the conclusion follows. Conversely, suppose . Then it follows that the point y must satisfy the inequality It follows that and y is indeed admissible, thereby ending the proof.
Next we show that there must exists some point in a bigger ball whose induced ball under compression has admissible points way off a certain line in the underlying ball. We find the following Lemma useful.   . This completes the proof of the theorem.

The needle function
In this section we introduce and study the needle function. We combine the geometry of lines under compression and the geometry of balls under compression to prove that this function is a function modeling an l-step self avoiding walk.
Definition 5.1. By the needle function of scale m and translation factor a, we mean the composite map . . , x n ) with x i = 0 for 1 ≤ i ≤ n and Γ a [ x] = (x 1 + a 1 , x 2 + a 2 , . . . , x n + a n ).
Proposition 5.1. The needle function Γ a • V m : R n −→ R n is a bijective map of order 2.
Proof. We remark that the translation with translation factor a for a fixed a given by Γ a : R n −→ R n is a bijective map. The result follows since the composite of bijective maps is still bijective.

A combinatorial interpretation
In this section we provide a combinatorial twist of the main result in this paper. We reformulate Theorem 5.2 in the language of graphs. We launch the following language: Definition 6.1 (Compression graphs). By a compression graph G of order k > 1 induced by x 1 = (u 1 , u 2 , . . . , u n ) ∈ R n with u i = 0 for all 1 ≤ i ≤ n, we mean the pair (V, E) where V is the vertex set and E the set of edges We now state a graph-theoretic version of Theorem 5.2.
Theorem 6.2 (A combinatorial version). There exists a compression graph of order l + 1 with l ∈ N and whose edges are paths.
Proof. Pick x 1 = (u 1 , u 2 , . . . , u n ) ∈ R n very far away from the origin with the property that u i > 1 and for all 1 ≤ i ≤ n. Next we apply the compression V 1 on where the points in set V are the vertices and each line in E are the edges of the graph G, with the edges being a path.