ROW_NUMBER function is nondeterministic

Question

Can we categorize ROW_NUMBER function as nondeterministic.  I think, unless we provide column names in the ORDER BY clause to uniquely identify a raw.

Answer 1

" windowed functions are non-deterministic either way you look at them ;)"
This is not true. Some Window Functions are deterministic others are not. The OP wanted to know if ROW_NUMBER was was non-deterministic; and yes, ROW_NUMBER is non-deterministic.
There are four functions in T-SQL that are referred to as "ranking functions": ROW_NUMBER, NTILE, RANK and DENSE_RANK. RANK and DENSE_RANK are deterministic, ROW_NUMBER and NTILE are not. To better understand this please note the queries below
(you can copy/paste into ssms): 
DECLARE @names TABLE (name_txt varchar(100) not null, phone_nbr varchar(20) not null);
WITH rand_names AS
SELECT TOP 10 * 
FROM (VALUES
('bob', '555-1212'),('bob', '555-1245'),('bob', '555-7896'),
('al', '555-8888'),('zach', '555-9999')
) t(n,p)
ORDER BY newid()
INSERT @names 
SELECT * FROM rand_names;
-- query1
SELECT [ROW_NUMBER()] = ROW_NUMBER() over (order by name_txt), 
[NTILE(2)] = NTILE(2) over (order by name_txt),
[RANK()] = RANK() over (order by name_txt),
[DENSE_RANK()] = DENSE_RANK() over (order by name_txt), * 
FROM @names;
-- query2
SELECT [ROW_NUMBER()] = ROW_NUMBER() over (order by (select null)), 
[NTILE(2)] = NTILE(2) over (order by (select null)),
[RANK()] = RANK() over (order by (select null)),
[DENSE_RANK()] = DENSE_RANK() over (order by (select null)), * 
FROM @names;